Continuity property of the set of real numbers. Axioms of real numbers

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✪ Axiom of continuity. Cantor's principle of nested cuts

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Axiom of continuity

The following proposition is perhaps the simplest and most convenient for applications formulation of the continuity property of real numbers. In the axiomatic construction of the theory of a real number, this statement, or equivalent to it, is certainly included in the number of axioms of a real number.

Axiom of continuity (completeness). A ⊂ R (\displaystyle A\subset \mathbb (R) ) and B ⊂ R (\displaystyle B\subset \mathbb (R) ) and the inequality is satisfied, there is such a real number ξ (\displaystyle \xi ) that for everyone a ∈ A (\displaystyle a\in A) and b ∈ B (\displaystyle b\in B) there is a relation

Geometrically, if we treat real numbers as points on a straight line, this statement seems obvious. If two sets A (\displaystyle A) and B (\displaystyle B) are such that on the number line all elements of one of them lie to the left of all elements of the second, then there is a number ξ (\displaystyle \xi ), separating these two sets, that is, lying to the right of all elements A (\displaystyle A)(except perhaps the ξ (\displaystyle \xi )) and to the left of all elements B (\displaystyle B)(same clause).

It should be noted here that despite the "obviousness" of this property, for rational numbers it is not always satisfied. For example, consider two sets:

A = ( x ∈ Q: x > 0 , x 2< 2 } , B = { x ∈ Q: x >0 , x 2 > 2 ) (\displaystyle A=$x\in \mathbb (Q) :x>0,\;x^(2)<2\},\quad B=\{x\in \mathbb {Q} :x>0,\;x^(2)>2$)

It is easy to see that for any elements a ∈ A (\displaystyle a\in A) and b ∈ B (\displaystyle b\in B) the inequality a< b {\displaystyle a. However rational numbers ξ (\displaystyle \xi ), separating these two sets, does not exist. Indeed, this number can only be 2 (\displaystyle (\sqrt (2))), but it is not rational.

The role of the axiom of continuity in the construction of mathematical analysis

The significance of the axiom of continuity is such that without it a rigorous construction of mathematical analysis is impossible. To illustrate, we present several fundamental statements of analysis, the proof of which is based on the continuity of real numbers:

(Theorem of Weierstrass). Every bounded monotonically increasing sequence converges
(Theorem Bolzano - Cauchy). A function continuous on a segment that takes on values of different signs at its ends vanishes at some interior point of the segment
(Existence of power, exponential, logarithmic and all trigonometric functions on the entire "natural" domain of definition). For example, it is proved that for every a > 0 (\displaystyle a>0) and whole n ⩾ 1 (\displaystyle n\geqslant 1) exist a n (\displaystyle (\sqrt[(n)](a))), that is, the solution of the equation x n = a , x > 0 (\displaystyle x^(n)=a,x>0). This allows you to determine the value of the expression for all rational x (\displaystyle x):

A m / n = (a n) m (\displaystyle a^(m/n)=\left((\sqrt[(n)](a))\right)^(m))

Finally, again due to the continuity of the number line, one can determine the value of the expression a x (\displaystyle a^(x)) already for arbitrary x ∈ R (\displaystyle x\in \mathbb (R) ). Similarly, using the continuity property, we prove the existence of the number log a ⁡ b (\displaystyle \log _(a)(b)) for any a , b > 0 , a ≠ 1 (\displaystyle a,b>0,a\neq 1).

For a long historical period of time, mathematicians proved theorems from analysis, in “thin places” referring to the geometric justification, and more often skipping them altogether, since it was obvious. The essential concept of continuity was used without any clear definition. It was only in the last third of the 19th century that the German mathematician Karl Weierstrass produced the arithmetization of analysis, constructing the first rigorous theory of real numbers as infinite decimal fractions. He proposed the classical definition of the limit in the language ε − δ (\displaystyle \varepsilon -\delta ), proved a number of statements that were considered “obvious” before him, and thus completed the construction of the foundation of mathematical analysis.

Later, other approaches to the definition of a real number were proposed. In the axiomatic approach, the continuity of real numbers is explicitly singled out as a separate axiom. In constructive approaches to the theory of a real number, for example, when constructing real numbers using Dedekind sections, the continuity property (in one formulation or another) is proved as a theorem.

Other Statements of the Continuity Property and Equivalent Propositions

There are several different statements expressing the continuity property of real numbers. Each of these principles can be taken as the basis for constructing the theory of the real number as an axiom of continuity, and all the others can be derived from it. This issue is discussed in more detail in the next section.

Continuity according to Dedekind

The question of the continuity of real numbers Dedekind considers in his work "Continuity and irrational numbers" . In it, he compares rational numbers with points on a straight line. As you know, between rational numbers and points of a straight line, you can establish a correspondence when the starting point and unit of measurement of the segments are chosen on the straight line. With the help of the latter, for every rational number a (\displaystyle a) construct the corresponding segment, and putting it aside to the right or to the left, depending on whether there is a (\displaystyle a) positive or negative number, get point p (\displaystyle p) corresponding to the number a (\displaystyle a). So every rational number a (\displaystyle a) matches one and only one point p (\displaystyle p) on a straight line.

It turns out that there are infinitely many points on the line that do not correspond to any rational number. For example, a point obtained by plotting the length of the diagonal of a square built on a unit segment. Thus, the realm of rational numbers does not have that completeness, or continuity, which is inherent in a straight line.

To find out what this continuity consists of, Dedekind makes the following remark. If a p (\displaystyle p) is a certain point of the line, then all points of the line fall into two classes: points located to the left p (\displaystyle p), and points to the right p (\displaystyle p). The very point p (\displaystyle p) can be arbitrarily assigned to either the lower or the upper class. Dedekind sees the essence of continuity in the reverse principle:

Geometrically, this principle seems obvious, but we are not in a position to prove it. Dedekind emphasizes that, in essence, this principle is a postulate, which expresses the essence of that property attributed to the direct line, which we call continuity.

To better understand the essence of the continuity of the number line in the sense of Dedekind, consider an arbitrary section of the set of real numbers, that is, the division of all real numbers into two non-empty classes, so that all numbers of one class lie on the number line to the left of all numbers of the second. These classes are named respectively lower and upper classes sections. Theoretically, there are 4 possibilities:

The lower class has a maximum element, the upper class does not have a minimum
The bottom class has no maximum element, while the top class has a minimum
The bottom class has a maximum element and the top class has a minimum element.
The bottom class has no maximum and the top class has no minimum.

In the first and second cases, the maximum element of the lower or the minimum element of the upper, respectively, produces this section. In the third case we have jump, and in the fourth space. Thus, the continuity of the number line means that there are no jumps or gaps in the set of real numbers, that is, figuratively speaking, there are no voids.

This proposition is also equivalent to Dedekind's continuity principle. Moreover, it can be shown that the assertion of the infimum theorem directly follows from the assertion of the supremum theorem, and vice versa (see below).

Finite cover lemma (Heine-Borel principle)

Finite Cover Lemma (Heine - Borel). In any system of intervals covering a segment, there is a finite subsystem covering this segment.

Limit point lemma (Bolzano-Weierstrass principle)

Limit Point Lemma (Bolzano - Weierstrass). Every infinite bounded number set has at least one limit point.. The second group expresses the fact that the set of real numbers is , and the order relation is consistent with the basic operations of the field. Thus, the first and second groups of axioms mean that the set of real numbers is an ordered field. The third group of axioms consists of one axiom - the axiom of continuity (or completeness).

To show the equivalence of different formulations of the continuity of the real numbers, one must prove that if one of these propositions holds for an ordered field, then all the others are true.

Theorem. Let be an arbitrary linear ordered set . The following statements are equivalent:

Whatever the non-empty sets and B ⊂ R (\displaystyle B\subset (\mathsf (R))), such that for any two elements a ∈ A (\displaystyle a\in A) and b ∈ B (\displaystyle b\in B) the inequality a ⩽ b (\displaystyle a\leqslant b), there is such an element ξ ∈ R (\displaystyle \xi \in (\mathsf (R))) that for everyone a ∈ A (\displaystyle a\in A) and b ∈ B (\displaystyle b\in B) there is a relation a ⩽ ξ ⩽ b (\displaystyle a\leqslant \xi \leqslant b)
For any section in R (\displaystyle (\mathsf (R))) there is an element that produces this section
Every non-empty set bounded above A ⊂ R (\displaystyle A\subset (\mathsf (R))) has a supremum
Every non-empty set bounded below A ⊂ R (\displaystyle A\subset (\mathsf (R))) has an infimum

As can be seen from this theorem, these four sentences use only what is on R (\displaystyle (\mathsf (R))) introduced a linear order relation, and do not use the field structure. Thus, each of them expresses the property R (\displaystyle (\mathsf (R))) as a linearly ordered set. This property (of an arbitrary linearly ordered set, not necessarily the set of real numbers) is called continuity, or completeness, according to Dedekind.

Proving the equivalence of other sentences already requires a field structure.

Theorem. Let be R (\displaystyle (\mathsf (R)))- an arbitrary ordered field. The following sentences are equivalent:

Comment. As can be seen from the theorem, the principle of nested segments in itself is not equivalent Dedekind's continuity principle. The principle of nested segments follows from the Dedekind continuity principle, but for the converse it is required to additionally require that the ordered field .

Plan:

1 Axiom of continuity
2 The role of the axiom of continuity in the construction of mathematical analysis
3 Other Statements of the Continuity Property and Equivalent Propositions
- 3.1 Continuity according to Dedekind
- 3.2 Lemma on nested segments (Cauchy-Cantor principle)
- 3.3 The supremum principle
- 3.4 Finite cover lemma (Heine-Borel principle)
- 3.5 Limit point lemma (Bolzano-Weierstrass principle)
4 Equivalence of sentences expressing the continuity of the set of real numbers

Introduction

Continuity of real numbers- a property of the system of real numbers, which the set of rational numbers does not have. Sometimes, instead of continuity, they talk about completeness of the system of real numbers. There are several different formulations of the continuity property, the best known of which are: Dedekind's principle of continuity of real numbers, principle of nested segments Cauchy - Cantor, supremum theorem. Depending on the accepted definition of a real number, the continuity property can either be postulated as an axiom - in one formulation or another, or proven as a theorem.

1. Axiom of continuity

Geometric illustration of the axiom of continuity

Axiom of continuity (completeness). Whatever the non-empty sets and , such that for any two elements and the inequality holds, there exists a number ξ such that for all and the relation holds

Geometrically, if we treat real numbers as points on a straight line, this statement seems obvious. If two sets A and B are such that on the number line all elements of one of them lie to the left of all elements of the second, then there is a number ξ, separating these two sets, that is, lying to the right of all elements A(except, perhaps, ξ itself) and to the left of all elements B(same clause).

It should be noted here that despite the "obviousness" of this property, for rational numbers it is not always satisfied. For example, consider two sets:

It is easy to see that for any elements and the inequality a < b. However rational there is no number ξ separating these two sets. Indeed, this number can only be , but it is not rational.

2. The role of the axiom of continuity in the construction of mathematical analysis

Finally, again due to the continuity of the number line, one can determine the value of the expression a x already for arbitrary . Similarly, using the continuity property, we prove the existence of the number log a b for any .

For a long historical period of time, mathematicians proved theorems from analysis, in “thin places” referring to geometric justification, and more often skipping them altogether because it was obvious. The essential concept of continuity was used without any clear definition. Only in the last third of the 19th century did the German mathematician Karl Weierstrass produce the arithmetization of analysis, constructing the first rigorous theory of real numbers as infinite decimal fractions. He proposed a classical definition of the limit in the language, proved a number of statements that were considered "obvious" before him, and thus completed the foundation of mathematical analysis.

Later, other approaches to the definition of a real number were proposed. In the axiomatic approach, the continuity of real numbers is explicitly singled out as a separate axiom. In constructive approaches to the theory of real numbers, for example, when constructing real numbers using Dedekind sections, the continuity property (in one formulation or another) is proved as a theorem.

3. Other formulations of the continuity property and equivalent propositions

3.1. Continuity according to Dedekind

The question of the continuity of real numbers is considered by Dedekind in his work Continuity and Irrational Numbers. In it, he compares the rational numbers with the points of a straight line. As is known, a correspondence can be established between rational numbers and points of a straight line when the starting point and the unit of measurement of the segments are chosen on the straight line. With the help of the latter, for every rational number a construct the corresponding segment, and putting it aside to the right or to the left, depending on whether there is a positive or negative number, get point p corresponding to the number a. So every rational number a matches one and only one point p on a straight line.

To find out what this continuity consists of, Dedekind makes the following remark. If a p is a certain point of the line, then all points of the line fall into two classes: points located to the left p, and points to the right p. The very point p can be arbitrarily assigned to either the lower or the upper class. Dedekind sees the essence of continuity in the reverse principle:

The bottom class has a maximum element, the top class does not have a minimum
The bottom class has no maximum element, while the top class has a minimum
The bottom class has a maximum element and the top class has a minimum element.
The bottom class has no maximum and the top class has no minimum.

If we introduce the concept of a section of the set of real numbers, then the Dedekind continuity principle can be formulated as follows.

Dedekind's continuity principle (completeness). For each section of the set of real numbers, there is a number that produces this section.

Comment. The formulation of the Axiom of Continuity about the existence of a point separating two sets is very reminiscent of the formulation of Dedekind's principle of continuity. In fact, these statements are equivalent, and, in essence, are different formulations of the same thing. Therefore, both of these statements are called the principle of continuity of real numbers according to Dedekind.

3.2. Lemma on nested segments (Cauchy-Cantor principle)

Lemma on nested segments (Cauchy - Kantor). Any system of nested segments

has a non-empty intersection, that is, there is at least one number that belongs to all segments of the given system.

If, in addition, the length of the segments of the given system tends to zero, that is,

then the intersection of the segments of this system consists of one point.

This property is called continuity of the set of real numbers in the sense of Cantor. It will be shown below that for the Archimedean ordered fields the continuity according to Cantor is equivalent to the continuity according to Dedekind.

3.3. The supremum principle

The supremacy principle. Every non-empty set of real numbers bounded above has a supremum.

In calculus courses, this proposition is usually a theorem, and its proof makes significant use of the continuity of the set of real numbers in one form or another. At the same time, on the contrary, it is possible to postulate the existence of a supremum for any non-empty set bounded from above, and relying on this to prove, for example, the Dedekind continuity principle. Thus, the supremum theorem is one of the equivalent formulations of the continuity property of real numbers.

Comment. Instead of the supremum, one can use the dual concept of the infimum.

The infimum principle. Every non-empty set of real numbers bounded below has an infimum.

3.4. Finite cover lemma (Heine-Borel principle)

Finite Cover Lemma (Heine - Borel). In any system of intervals covering a segment, there is a finite subsystem covering this segment.

3.5. Limit point lemma (Bolzano-Weierstrass principle)

Limit Point Lemma (Bolzano - Weierstrass). Every infinite bounded number set has at least one limit point.

4. Equivalence of sentences expressing the continuity of the set of real numbers

Let's make some preliminary remarks. In accordance with the axiomatic definition of a real number, the set of real numbers satisfies three groups of axioms. The first group is the field axioms. The second group expresses the fact that the collection of real numbers is a linearly ordered set, and the order relation is consistent with the basic operations of the field. Thus, the first and second groups of axioms mean that the set of real numbers is an ordered field. The third group of axioms consists of one axiom - the axiom of continuity (or completeness).

To show the equivalence of different formulations of the continuity of the real numbers, one must prove that if one of these propositions holds for an ordered field, then all the others are true.

Theorem. Let be an arbitrary linearly ordered set. The following statements are equivalent:

As can be seen from this theorem, these four propositions only use what the linear order relation has introduced and do not use the field structure. Thus, each of them expresses a property as a linearly ordered set. This property (of an arbitrary linearly ordered set, not necessarily the set of real numbers) is called continuity, or completeness, according to Dedekind.

Proving the equivalence of other sentences already requires a field structure.

Theorem. Let be an arbitrary ordered field. The following sentences are equivalent:

The proof of the above theorems can be found in the books from the bibliography given below.

Notes

Zorich, V. A. Mathematical analysis. Part I. - Ed. 4th, corrected .. - M .: "MTsNMO", 2002. - S. 43.
For example, in the axiomatic definition of a real number, the Dedekind continuity principle is included among the axioms, and in the constructive definition of a real number using Dedekind sections, the same statement is already a theorem - see for example Fikhtengolts, G. M.
Kudryavtsev, L. D. Course of mathematical analysis. - 5th ed. - M .: "Drofa", 2003. - T. 1. - S. 38.
Kudryavtsev, L. D. Course of mathematical analysis. - 5th ed. - M .: "Drofa", 2003. - T. 1. - S. 84.
Zorich, V. A. Mathematical analysis. Part I. - Ed. 4th, corrected .. - M .: "MTsNMO", 2002. - S. 81.
Dedekind, R. Continuity and irrational numbers - www.mathesis.ru/book/dedekind4 = Stetigkeit und irrationale Zahlen. - 4th revised edition. - Odessa: Mathesis, 1923. - 44 p.

Literature

Kudryavtsev, L. D. Course of mathematical analysis. - 5th ed. - M .: "Drofa", 2003. - T. 1. - 704 p. - ISBN 5-7107-4119-1
Fikhtengolts, G. M. Fundamentals of mathematical analysis. - 7th ed. - M .: "FIZMATLIT", 2002. - T. 1. - 416 p. - ISBN 5-9221-0196-X
Dedekind, R. Continuity and irrational numbers - www.mathesis.ru/book/dedekind4 = Stetigkeit und irrationale Zahlen. - 4th revised edition. - Odessa: Mathesis, 1923. - 44 p. , Turing completeness , Set partition , Set variation , Set degree .

Definition of nested segments. Proof of the Cauchy-Cantor lemma on nested segments.

Content

Definition of nested segments

Let a and b be two real numbers (). Let it go . The set of numbers x satisfying the inequalities is called a segment with ends a and b . The segment is marked like this:

Sequence of number segments

called the sequence nested segments, if each subsequent segment is contained in the previous one:
.
That is, the ends of the segments are connected by inequalities:
.

Lemma on nested segments (Cauchy-Cantor principle)

For any sequence of nested segments, there exists a point that belongs to all these segments.
If the lengths of the segments tend to zero:
,
then there is only one such point.

This lemma is also called nested segment theorem or Cauchy-Cantor principle.

Proof

For proof the first part of the lemma, we use the axiom of the completeness of real numbers.

Axiom of completeness of real numbers is as follows. Let sets A and B be two subsets of real numbers such that the inequality holds for any two elements and these sets. Then there is a real number c such that for all and the inequalities hold:
.

Let's apply this axiom. Let the set A be the set of left ends of the segments, and the set B be the set of right ends. Then the inequality holds between any two elements of these sets. Then it follows from the axiom of completeness of real numbers that there is such a number c that for all n the following inequalities hold:
.
It means that point c belongs to all segments.

Let's prove the second part of the lemma.

Let be . According to the definition of the limit of a sequence, this means that for any positive number there exists a natural number N that depends on ε such that for all natural numbers n > N the inequality
(1) .

Let's assume the opposite. Let there be two distinct points c 1 and c 2 , c 1 ≠ c2 belonging to all segments. This means that the following inequalities hold for all n:
;
.
From here
.
Applying (1) we have:
.
This inequality must hold for any positive values of ε. Hence it follows that
c 1 = c2.

The lemma is proven.

Comment

The existence of a point belonging to all segments follows from the axiom of completeness, which is valid for real numbers. This axiom does not apply to rational numbers. Therefore, the lemma on nested segments also does not apply to the set of rational numbers.

For example, we could choose the segments so that both the left and right ends converge to an irrational number . Then any rational number, with an increase in n, would always fall out of the system of segments. The only number that belongs to all the segment is an irrational number.

References:
O.V. Demons. Lectures on mathematical analysis. Part 1. Moscow, 2004.

In the school mathematics course, real numbers were determined in a constructive way, based on the need to make measurements. Such a definition was not rigorous and often led researchers into a dead end. For example, the question of the continuity of real numbers, that is, whether there are voids in this set. Therefore, when conducting mathematical research, it is necessary to have a strict definition of the concepts under study, at least within the framework of some intuitive assumptions (axioms) that are consistent with practice.

Definition. Set of elements x, y, z, …, consisting of more than one element, is called a set R real numbers, if the following operations and relations are established for these objects:

I group of axioms are the axioms of the addition operation.

in multitude R the addition operation is introduced, that is, for any pair of elements a and b sum and denoted a + b
I 1 . a+b=b+a, a, b R .

I 2 . a+(b+c)=(a+b)+c,a, b, c R .

I 3. There is such an element called zero and denoted by 0, which for any a R the condition a+0=a.

I 4 . For any element a R there is an element called him opposite and denoted - a, for which a+(-a)=0. Element a+(-b), a, b R , is called difference elements a and b and denoted a - b.

II – group of axioms - axioms of the operation of multiplication. in multitude R operation entered multiplication, that is, for any pair of elements a and b a single element is defined, called them work and denoted a b, so that the following conditions are satisfied:
II 1 . ab=ba, a, b R .

II 2 a(bc)=(ab)c, a, b, c R .

II 3 . There is an element called unit and denoted by 1, which for any a R the condition a 1=a.

II 4 . For anyone a 0 there is an element called him reverse and denoted by or 1/ a, for which a=1. Element a , b 0, called private from division a on the b and denoted a:b or or a/b.

II 5 . Relationship between addition and multiplication operations: for any a, b, c R the condition is met ( ac+b)c=ac+bc.

A set of objects that satisfies the axioms of groups I and II is called a numerical field or simply a field. And the corresponding axioms are called field axioms.

III - the third group of axioms - axioms of order. For elements R order relation is defined. It consists of the following. For any two different elements a and b one of two relations holds: either a b(read " a less or equal b"), or a b(read " a more or equal b"). This assumes that the following conditions are met:

III 1. a a for everybody a. From a b, b should a=b.

III 2 . Transitivity. If a a b and b c, then a c.

III 3 . If a a b, then for any element c takes place a+c b+c.

III 4 . If a a 0,b 0, then ab 0 .

IV group of axioms consists of one axiom - the axiom of continuity. For any non-empty sets X and Y from R such that for each pair of elements x X and y Y the inequality x < y, there is an element a R, satisfying the condition

Rice. 2

x < a < y, x X, y Y(Fig. 2). The enumerated properties completely define the set of real numbers in the sense that all its other properties follow from these properties. This definition uniquely defines the set of real numbers up to the specific nature of its elements. The caveat that a set contains more than one element is necessary because a set consisting of zero alone satisfies all the axioms in an obvious way. In what follows, elements of the set R will be called numbers.

Let us now define the familiar concepts of natural, rational and irrational numbers. The numbers 1, 2 1+1, 3 2+1, ... are called natural numbers, and their set is denoted N . From the definition of the set of natural numbers it follows that it has the following characteristic property: if

1) A N ,

3) for each element x A the inclusion x+ 1 A, then A=N .

Indeed, according to condition 2) we have 1 A, therefore, by property 3) and 2 A, and then, according to the same property, we obtain 3 A. Since any natural number n is obtained from 1 by successively adding the same 1 to it, then n A, i.e. N A, and since condition 1 satisfies the inclusion A N , then A=N .

The proof principle is based on this property of natural numbers. by mathematical induction. If there are many statements, each of which is assigned a natural number (its number) n=1, 2, ..., and if it is proved that:

1) the statement with number 1 is true;

2) from the validity of the statement with any number n N follows the validity of the statement with the number n+1;

then the validity of all statements is proved, i.e., any statement with an arbitrary number n N .

Numbers 0, + 1, + 2, ... called whole numbers, their set is denoted Z .

Type numbers m/n, where m and n whole, and n 0 are called rational numbers. The set of all rational numbers is denoted Q .

Real numbers that are not rational are called irrational, their set is denoted I .

The question arises that perhaps the rational numbers exhaust all the elements of the set R? The answer to this question is given by the axiom of continuity. Indeed, this axiom does not hold for rational numbers. For example, consider two sets:

It is easy to see that for any elements and the inequality is fulfilled. However rational there is no number separating these two sets. Indeed, this number can only be , but it is not rational. This fact indicates that there are irrational numbers in the set R.

In addition to the four arithmetic operations on numbers, you can perform exponentiation and root extraction. For any number a R and natural n degree a n defined as a product n factors equal to a:

A-priory a 0 1, a>0, a-n 1/ a n a 0, n- natural number.

Example. Bernoulli's inequality: ( 1+x)n> 1+nx Prove by induction.

Let be a>0, n- natural number. Number b called root n th degree from among a, if b n =a. In this case, it is written Existence and uniqueness of a positive root of any degree n from any positive number will be proved below in § 7.3.
Even root, a 0 has two meanings: if b = , k N , then and -b= . Indeed, from b 2k = a follows that

(-b)2k = ((-b) 2 )k = (b 2)k = b 2k

A non-negative value is called its arithmetic value.
If a r = p/q, where p and q whole, q 0, i.e. r is a rational number, then a > 0

(2.1)

So the degree a r defined for any rational number r. It follows from its definition that for any rational r there is an equality

a -r = 1/a r.

Degree a x(number x called exponent) for any real number x is obtained by extending the degree continuously with a rational exponent (see Section 8.2 for more on this). For any number a R non-negative number

called him absolute value or module. For the absolute values of numbers, the inequalities

|a + b| < |a| + |b|,
||a - b|| < |a - b|, a, b R

They are proved using properties I-IV of the real numbers.

The role of the axiom of continuity in the construction of mathematical analysis

The significance of the axiom of continuity is such that without it a rigorous construction of mathematical analysis is impossible. [ source unspecified 1351 days] To illustrate, we present several fundamental statements of analysis, the proof of which is based on the continuity of real numbers:

· (Weierstrass theorem). Every bounded monotonically increasing sequence converges

· (Bolzano-Cauchy theorem). A function continuous on a segment that takes on values of different signs at its ends vanishes at some interior point of the segment

· (Existence of power, exponential, logarithmic and all trigonometric functions on the entire "natural" domain of definition). For example, it is proved that for every integer there exists , that is, a solution to the equation . This allows you to determine the value of the expression for all rational :

Finally, again, due to the continuity of the number line, it is possible to determine the value of the expression already for an arbitrary . Similarly, using the continuity property, we prove the existence of a number for any .

Later, other approaches to the definition of a real number were proposed. In the axiomatic approach, the continuity of real numbers is explicitly singled out as a separate axiom. In constructive approaches to the theory of real numbers, for example, when constructing real numbers using Dedekind sections, the continuity property (in one formulation or another) is proved as a theorem.

Other formulations of the continuity property and equivalent sentences[edit | edit wiki text]

Continuity according to Dedekind[edit | edit wiki text]

Main article:Section theory in the region of rational numbers

The question of the continuity of real numbers is considered by Dedekind in his work Continuity and Irrational Numbers. In it, he compares the rational numbers with the points of a straight line. As you know, between rational numbers and points of a straight line, you can establish a correspondence when you choose the starting point and the unit of measurement of the segments on the straight line. With the help of the latter, it is possible to construct the corresponding segment for each rational number, and putting it aside to the right or to the left, depending on whether there is a positive or negative number, get a point corresponding to the number. Thus, each rational number corresponds to one and only one point on the line.

To find out what this continuity consists of, Dedekind makes the following remark. If there is a certain point of the line, then all points of the line fall into two classes: points located to the left, and points located to the right. The point itself can be arbitrarily assigned either to the lower or to the upper class. Dedekind sees the essence of continuity in the reverse principle:

1. The lower class has a maximum element, the upper class does not have a minimum

2. There is no maximum element in the lower class, while there is a minimum element in the upper class

3. The bottom class has the maximum and the top class has the minimum elements

4. There is no maximum element in the lower class, and there is no minimum element in the upper class

If we introduce the concept of a section of the set of real numbers, then the Dedekind continuity principle can be formulated as follows.

Dedekind's continuity principle (completeness). For each section of the set of real numbers, there is a number that produces this section.

Lemma on nested segments (Cauchy-Cantor principle)[edit | edit wiki text]

Main article:Lemma on nested segments

Lemma on nested segments (Cauchy - Kantor). Any system of nested segments

has a non-empty intersection, that is, there is at least one number that belongs to all segments of the given system.

If, in addition, the length of the segments of the given system tends to zero, that is,

then the intersection of the segments of this system consists of one point.

This property is called continuity of the set of real numbers in the sense of Cantor. It will be shown below that for Archimedean ordered fields, Cantor continuity is equivalent to Dedekind continuity.

The supremum principle[edit | edit wiki text]

The supremacy principle. Every non-empty set of real numbers bounded above has a supremum.

Comment. Instead of the supremum, one can use the dual concept of the infimum.

The infimum principle. Every non-empty set of real numbers bounded below has an infimum.

Finite cover lemma (Heine-Borel principle)[edit | edit wiki text]

Main article:Heine-Borel Lemma

Finite Cover Lemma (Heine - Borel). In any system of intervals covering a segment, there is a finite subsystem covering this segment.

Limit point lemma (Bolzano-Weierstrass principle)[edit | edit wiki text]

Main article:Bolzano-Weierstrass theorem

Limit Point Lemma (Bolzano - Weierstrass). Every infinite bounded number set has at least one limit point.

Equivalence of sentences expressing the continuity of the set of real numbers[edit | edit wiki text]

To show the equivalence of different formulations of the continuity of the real numbers, one must prove that if one of these propositions holds for an ordered field, then all the others are true.

Theorem. Let be an arbitrary linearly ordered set. The following statements are equivalent:

1. Whatever non-empty sets and are such that for any two elements and , there exists an element such that for all and , the relation holds

2. For any section in there exists an element that produces this section

3. Every non-empty set bounded above has a supremum

4. Every non-empty set bounded below has an infimum

Proving the equivalence of other sentences already requires a field structure.

Theorem. Let be an arbitrary ordered field. The following sentences are equivalent:

1. (as a linearly ordered set) is Dedekind complete

2. To fulfilled the principle of Archimedes and principle of nested segments

3. For the Heine-Borel principle is fulfilled

4. For the Bolzano-Weierstrass principle is fulfilled

The proof of the above theorems can be found in the books from the bibliography given below.

· Kudryavtsev, L. D. Course of mathematical analysis. - 5th ed. - M.: "Drofa", 2003. - T. 1. - 704 p. - ISBN 5-7107-4119-1.

· Fikhtengolts, G. M. Fundamentals of mathematical analysis. - 7th ed. - M.: "FIZMATLIT", 2002. - T. 1. - 416 p. - ISBN 5-9221-0196-X.

· Dedekind, R. Continuity and irrational numbers = Stetigkeit und irrationale Zahlen. - 4th revised edition. - Odessa: Mathesis, 1923. - 44 p.

· Zorich, V. A. Mathematical analysis. Part I. - Ed. 4th, corrected .. - M .: "MTsNMO", 2002. - 657 p. - ISBN 5-94057-056-9.

· Continuity of functions and numerical domains: B. Bolzano, L. O. Cauchy, R. Dedekind, G. Kantor. - 3rd ed. - Novosibirsk: ANT, 2005. - 64 p.

4.5. Axiom of continuity

Whatever two non-empty sets of real numbers A and

B , for which, for any elements a ∈ A and b ∈ B, the inequality

a ≤ b , there exists a number λ such that for all a ∈ A , b ∈ B

equality a ≤ λ ≤ b .

The continuity property of real numbers means that on the real

there are no “voids” on the vein line, that is, points representing numbers fill

the entire real axis.

Let us give another formulation of the axiom of continuity. For this we introduce

Definition 1.4.5. Two sets A and B will be called a section

sets of real numbers, if

1) the sets A and B are not empty;

2) the union of the sets A and B constitutes the set of all real

numbers;

3) each number of set A is less than the number of set B .

That is, each set forming a section contains at least one

element, these sets do not contain common elements and, if a ∈ A and b ∈ B , then

The set A will be called the lower class, and the set B will be called the upper class.

section class. We will designate the section as A B .

The simplest examples of sections are the sections obtained as follows.

blowing way. Take some number α and set

A = ( x x< α } , B = { x x ≥ α } . Легко видеть, что эти множества не пусты, не пере-

intersect and if a ∈ A and b ∈ B , then a< b , поэтому множества A и B образуют

section. Similarly, one can form a section, by sets

A =(x x ≤ α ) , B =(x x > α ) .

Such sections will be called sections generated by the number α or

we will say that the number α produces this section. This can be written as

Sections generated by any number have two interesting

properties:

Property 1. Either the upper class contains the smallest number, and in the lower

class does not have the largest number, or the lower class contains the largest number

lo, and the top class is not the smallest.

Property 2. The number generating the given section is unique.

It turns out that the continuity axiom formulated above is equivalent to

is consistent with the statement called Dedekind's principle:

Dedekind principle. For each section, there is a number generating

this is a section.

Let us prove the equivalence of these statements.

Let the axiom of continuity be valid, and some se-

value A B . Then, since the classes A and B satisfy the conditions, the formulas

axiom, there exists a number λ such that a ≤ λ ≤ b for any numbers

a ∈ A and b ∈ B . But the number λ must belong to one and only one of

classes A or B , so one of the inequalities a ≤ λ< b или

a< λ ≤ b . Таким образом, число λ либо является наибольшим в нижнем классе,

or the smallest in the upper class and generates the given section.

Conversely, let the Dedekind principle be satisfied and two non-empty

sets A and B such that for all a ∈ A and b ∈ B the inequality

a ≤ b . Denote by B the set of numbers b such that a ≤ b for any

b ∈ B and all a ∈ A . Then B ⊂ B . For the set A we take the set of all numbers

villages not included in B .

Let us prove that the sets A and B form a section.

Indeed, it is obvious that the set B is not empty, since it contains

non-empty set B . The set A is not empty either, because if a number a ∈ A ,

then the number a − 1∉ B , since any number included in B must be at least

numbers a , hence a − 1∈ A .

the set of all real numbers, by virtue of the choice of sets.

And finally, if a ∈ A and b ∈ B , then a ≤ b . Indeed, if any

number c satisfies the inequality c > b , where b ∈ B , then the false

the equality c > a (a is an arbitrary element of the set A) and c ∈ B .

So, A and B form a section, and by virtue of the Dedekind principle, there is a number

lo λ , generating this section, that is, which is either the largest in the class

Let us prove that this number cannot belong to the class A . Valid-

but if λ ∈ A , then there is a number a* ∈ A such that λ< a* . Тогда существует

the number a′ lying between the numbers λ and a* . From the inequality a′< a* следует, что

a′ ∈ A , then from the inequality λ< a′ следует, что λ не является наибольшим в

class A , which contradicts the Dedekind principle. Therefore, the number λ will

is the smallest in the class B and for all a ∈ A and the inequality

a ≤ λ ≤ b , as required.◄

Thus, the property formulated in the axiom and the property,

formulated in the Dedekind principle are equivalent. In the future, these

properties of the set of real numbers we will call continuity

according to Dedekind.

The continuity of the set of real numbers according to Dedekind implies

two important theorems.

Theorem 1.4.3. (Archimedes principle) Whatever the real number

a, there is a natural number n such that a< n .

Let us assume that the statement of the theorem is false, that is, there exists such

some number b0 such that the inequality n ≤ b0 holds for all natural numbers

n. Let us divide the set of real numbers into two classes: in the class B we assign

all numbers b that satisfy the inequality n ≤ b for any natural n .

This class is not empty, since the number b0 belongs to it. We assign everything to class A

the remaining numbers. This class is also not empty, since any natural number

is included in A . Classes A and B do not intersect and their union is

the set of all real numbers.

If we take arbitrary numbers a ∈ A and b ∈ B , then there is a natural

number n0 such that a< n0 ≤ b , откуда следует, что a < b . Следовательно, классы

A and B satisfy the Dedekind principle and there is a number α that

generates a section A B , that is, α is either the largest in the class A , or

bo the smallest in class B . If we assume that α belongs to the class A , then

one can find a natural number n1 for which the inequality α< n1 .

Since n1 is also included in A , the number α will not be the largest in this class,

therefore, our assumption is wrong and α is the smallest in

class B.

On the other hand, take a number α − 1 that belongs to the class A . Follow-

Therefore, there is a natural number n2 such that α − 1< n2 , откуда получим

α < n2 + 1 . Так как n2 + 1 - натуральное число, то из последнего неравенства

it follows that α ∈ A . The resulting contradiction proves the theorem.◄

Consequence. Whatever the numbers a and b are such that 0< a < b , существует

a natural number n for which the inequality na > b holds.

To prove it, it suffices to apply the principle of Archimedes to the number

and use the property of inequalities.◄

The corollary has a simple geometric meaning: Whatever the two

segment, if on the larger of them, from one of its ends successively

put a smaller one, then in a finite number of steps it is possible to go beyond

larger cut.

Example 1. Prove that for every non-negative number a there exists

the only non-negative real number t such that

t n = a, n ∈ , n ≥ 2 .

This theorem on the existence of an arithmetic root of the nth degree

from a non-negative number in the school course of algebra is accepted without proof

pledges.

☺If a = 0 , then x = 0 , so the proof of the existence of arithmetic

The root of a is only required for a > 0 .

Assume that a > 0 and partition the set of all real numbers

for two classes. We assign to class B all positive numbers x that satisfy

create the inequality x n > a , into the class A , all the rest.

According to the axiom of Archimedes, there are natural numbers k and m such that

< a < k . Тогда k 2 ≥ k >a and 2 ≤< a , т.е. оба класса непусты, причем класс

A contains positive numbers.

Obviously, A ∪ B = and if x1 ∈ A and x2 ∈ B , then x1< x2 .

Thus the classes A and B form a section. The number that makes up this

section, denoted by t . Then t is either the largest number in the class

all A , or the smallest in class B .

Assume that t ∈ A and t n< a . Возьмем число h , удовлетворяющее нера-

0< h < 1 . Тогда

(t + h)n = t n + Cnt n−1h + Cn t n−2h2 + ... + Cnn hn< t n + Cnt n−1h + Cn t n−2h + ... + Cn h =

T n + h (Cnt n−1 + Cn t n−2 + ... + Cn + Cn t n) − hCn t n = t n + h (t + 1) − ht n =

T n + h (t + 1) − t n

Then we get (t + h)< a . Это означает,

Hence, if we take h<

that t + h ∈ A , which contradicts the fact that t is the largest element in the class A .

Similarly, if we assume that t is the smallest element of class B,

then, taking a number h that satisfies the inequalities 0< h < 1 и h < ,

we get (t − h) = t n − Cnt n−1h + Cn t n−2 h 2 − ... + (−1) Cn h n >

> t n − Cnt n−1h + Cn t n−2h + ... + Cn h = t n − h (t + 1) − t n > a .

This means that t − h ∈ B and t cannot be the smallest element

class B. Therefore, t n = a .

Uniqueness follows from the fact that if t1< t2 , то t1n < t2 .☻ n

Example 2. Prove that if a< b , то всегда найдется рациональное число r

such that a< r < b .

☺If the numbers a and b are rational, then the number is rational and

satisfies the required conditions. Suppose that at least one of the numbers a or b

irrational, for example, let's say that the number b is irrational. Assumed

we also press that a ≥ 0 , then b > 0 . We write the representations of the numbers a and b in the form

decimal fractions: a = α 0 ,α1α 2α 3.... and b = β 0 , β1β 2 β3... , where the second fraction is infinite

finite and non-periodic. As for the representation of the number a, then we will count

that if the number a is rational, then its notation is either finite or it is

rhyonic fraction whose period is not equal to 9.

Since b > a , then β 0 ≥ α 0 ; if β 0 = α 0 , then β1 ≥ α1 ; if β1 = α1 , then β 2 ≥ α 2

etc., and there is such a value i , at which for the first time it will be

satisfy the strict inequality βi > α i . Then the number β 0 , β1β 2 ...βi will be rational

real and will lie between the numbers a and b.

If a< 0 , то приведенное рассуждение надо применить к числам a + n и

b + n, where n is a natural number such that n ≥ a. The existence of such a number

follows from the axiom of Archimedes. ☻

Definition 1.4.6. Let a sequence of segments of the real axis be given

([ an ; bn ]) , an< bn . Эту последовательность будем называть системой вло-

intervals if for any n the inequalities an ≤ an+1 hold and

For such a system, the inclusions

[a1; b1 ] ⊃ [ a2 ; b2 ] ⊃ [ a3 ; b3] ⊃ ... ⊃ [ an ; bn] ⊃ ... ,

that is, each next segment is contained in the previous one.

Theorem 1.4.4. For any system of nested segments, there exists

at least one point that is included in each of these segments.

Let's take two sets A = (an ) and B = (bn ) . They are not empty and for any

n and m, the inequality an< bm . Докажем это.

If n ≥ m , then an< bn ≤ bm . Если n < m , то an ≤ am < bm .

Thus the classes A and B satisfy the axiom of continuity and,

therefore, there exists a number λ such that an ≤ λ ≤ bn for any n, i.e. This

the number belongs to any segment [ an ; bn] .◄

In what follows (Theorem 2.1.8), we refine this theorem.

The statement formulated in Theorem 1.4.4 is called the principle

Cantor, and the set that satisfies this condition will be called

discontinuous according to Cantor.

We have proved that if an ordered set is Dede-continuous

kindu, then the principle of Archimedes is fulfilled in it and it is continuous according to Cantor.

It can be proved that an ordered set in which the principles

principles of Archimedes and Cantor will be continuous according to Dedekind. Proof

this fact is contained, for example, in .

The principle of Archimedes allows each segment of a straight line to compare

which is the only positive number that satisfies the conditions:

1. equal segments correspond to equal numbers;

2. If the point of the segment AC and the segments AB and BC correspond to the numbers a and

b, then the segment AC corresponds to the number a + b;

3. a certain segment corresponds to the number 1.

The number corresponding to each segment and satisfying the conditions 1-3 on-

is called the length of this segment.

Cantor's principle allows us to prove that for every positive

number, you can find a segment whose length is equal to this number. Thus,

between the set of positive real numbers and the set of segments

kov, which are laid off from some point of the straight line on a given side

from this point, a one-to-one correspondence can be established.

This allows us to define the numerical axis and introduce a correspondence between the

waiting for real numbers and points on the line. To do this, let's take some

I draw a line and choose a point O on it, which divides this line into two

beam. We call one of these rays positive, and the second negative.

nym. Then we will say that we have chosen the direction on this straight line.

Definition 1.4.7. The real axis is the straight line on which

a) point O, called the origin or origin;

b) direction;

c) a segment of unit length.

Now, to each real number a, we associate a point M on the number

howl straight so that

a) the number 0 corresponded to the origin;

b) OM = a - the length of the segment from the origin to the point M was equal to

modulo number;

c) if a is positive, then the point is taken on the positive ray and, es-

If it is negative, then it is negative.

This rule establishes a one-to-one correspondence between

the set of real numbers and the set of points on the line.

The number line (axis) will also be called the real line

This also implies the geometric meaning of the modulus of a real number.

la: the modulus of the number is equal to the distance from the origin to the point depicted

plotting this number on the number line.

We can now give a geometric interpretation to properties 6 and 7

modulus of a real number. With a positive C of the number x, satisfy-

properties 6 fill the interval (−C , C) , and the numbers x satisfying

property 7 lie on the rays (−∞,C) or (C , +∞) .

We note one more remarkable geometric property of the real module.

real number.

The modulus of the difference of two numbers is equal to the distance between the points, respectively

corresponding to these numbers on the real axis.

ry standard numerical sets.

The set of natural numbers;

Set of integers;

The set of rational numbers;

The set of real numbers;

Sets, respectively, of integers, rational and real

real non-negative numbers;

Set of complex numbers.

In addition, the set of real numbers is denoted as (−∞, +∞) .

Subsets of this set:

(a, b) = ( x | x ∈ R, a< x < b} - интервал;

[ a, b] = ( x | x ∈ R, a ≤ x ≤ b) - segment;

(a, b] = ( x | x ∈ R, a< x ≤ b} или [ a, b) = { x | x ∈ R, a ≤ x < b} - полуинтерва-

ly or half-segments;

(a, +∞) = ( x | x ∈ R, a< x} или (−∞, b) = { x | x ∈ R, x < b} - открытые лучи;

[ a, +∞) = ( x | x ∈ R, a ≤ x) or (−∞, b] = ( x | x ∈ R, x ≤ b) are closed rays.

Finally, sometimes we will need gaps in which we will not care

whether its ends belong to this interval or not. Such a gap will

denote a, b.

§ 5 Boundedness of numerical sets

Definition 1.5.1. The number set X is called bounded

from above if there exists a number M such that x ≤ M for any element x from

sets X .

Definition 1.5.2. The number set X is called bounded

from below if there exists a number m such that x ≥ m for any element x from

sets X .

Definition 1.5.3. The number set X is called bounded,

if it is bounded from above and below.

In symbolic notation, these definitions will look like this:

a set X is bounded from above if ∃M ∀x ∈ X: x ≤ M ,

bounded from below if ∃m ∀x ∈ X: x ≥ m and

is bounded if ∃m, M ∀x ∈ X: m ≤ x ≤ M .

Theorem 1.5.1. A number set X is bounded if and only if

when there is a number C such that for all elements x from this set

, the inequality x ≤ C is satisfied.

Let the set X be bounded. We put C \u003d max (m, M) - the most

the greater of the numbers m and M . Then, using the properties of the modulus of real

numbers, we obtain the inequalities x ≤ M ≤ M ≤ C and x ≥ m ≥ − m ≥ −C , whence

not that x ≤ C .

Conversely, if x ≤ C , then −C ≤ x ≤ C . This is the tre-

given if we set M = C and m = −C .◄

The number M that bounds the set X from above is called the upper

set boundary. If M is the upper bound of a set X, then any

the number M ′ , which is greater than M , will also be the upper bound of this set.

Thus, we can talk about the set of upper bounds of the set

x. Denote the set of upper bounds by M . Then, ∀x ∈ X and ∀M ∈ M

the inequality x ≤ M will be satisfied, therefore, according to the axiom, continuously

There exists a number M 0 such that x ≤ M 0 ≤ M . This number is called the

the upper bound of the number set X or the upper bound of this

set or the supremum of the set X and is denoted by M 0 = sup X .

Thus, we have proved that every non-empty numerical set,

bounded above always has an exact upper bound.

Obviously, the equality M 0 = sup X is equivalent to two conditions:

1) ∀x ∈ X, x ≤ M 0 , i.e., M 0 - the upper limit of the set

2) ∀ε > 0 ∃xε ∈ X so that xε > M 0 − ε , i.e., this gra-

nitsa cannot be improved (reduced).

Example 1. Consider the set X = ⎨1 − ⎬ . Let us prove that sup X = 1 .

☺Indeed, firstly, the inequality 1 −< 1 выполняется для любого

n ∈ ; secondly, if we take an arbitrary positive number ε, then by

the principle of Archimedes, one can find a natural number nε such that nε > . That-

when the inequality 1 − > 1 − ε is satisfied, i.e., found an element xnε of the

of X greater than 1 − ε , which means that 1 is the least upper bound

Similarly, one can prove that if a set is bounded below, then

it has a sharp lower bound, which is also called the lower bound.

the new or infimum of the set X and is denoted by inf X .

The equality m0 = inf X is equivalent to the conditions:

1) ∀x ∈ X the inequality x ≥ m0 holds;

2) ∀ε > 0 ∃xε ∈ X so that the inequality xε< m0 + ε .

If the set X has the largest element x0 , then we will call it

the maximum element of the set X and denote x0 = max X . Then

sup X = x0 . Similarly, if there is a smallest element in a set, then

we will call it minimal, denote min X and it will be in-

phimum of the set X .

For example, the set of natural numbers has the smallest element -

unit, which is also the infimum of the set. Super-

mum does not have this set, since it is not bounded from above.

The definitions of precise upper and lower bounds can be extended to

sets unbounded from above or below, setting sup X = +∞ or, respectively,

Correspondingly, inf X = −∞ .

In conclusion, we formulate several properties of upper and lower bounds.

Property 1. Let X be some numerical set. Denote by

− X set (− x | x ∈ X ) . Then sup (− X) = − inf X and inf (− X) = − sup X .

Property 2. Let X be some numerical set λ - real

number. Denote by λ X the set (λ x | x ∈ X ) . Then if λ ≥ 0, then

sup (λ X) = λ sup X , inf (λ X) = λ inf X and, if λ< 0, то

sup (λ X) = λ inf X , inf (λ X) = λ sup X .

Property 3. Let X1 and X 2 be numerical sets. Denote by

X1 + X 2 set ( x1 + x2 | x1 ∈ X 1, x2 ∈ X 2 ) and through X1 − X 2 the set

( x1 − x2 | x1 ∈ X1, x2 ∈ X 2) . Then sup (X 1 + X 2) = sup X 1 + sup X 2 ,

inf (X1 + X 2) = inf X1 + inf X 2 , sup (X 1 − X 2) = sup X 1 − inf X 2 and

inf (X1 − X 2) = inf X1 − sup X 2 .

Property 4. Let X1 and X 2 be numerical sets, all elements of which

ryh are non-negative. Then

sup (X1 X 2) = sup X1 ⋅ sup X 2 , inf (X1 X 2) = inf X 1 ⋅ inf X 2 .

Let us prove, for example, the first equality in property 3.

Let x1 ∈ X1, x2 ∈ X 2 and x = x1 + x2 . Then x1 ≤ sup X1, x2 ≤ sup X 2 and

x ≤ sup X1 + sup X 2 , whence sup (X1 + X 2) ≤ sup X1 + sup X 2 .

To prove the opposite inequality, take the number

y< sup X 1 + sup X 2 . Тогда можно найти элементы x1 ∈ X1 и x2 ∈ X 2 такие,

what x1< sup X1 и x2 < sup X 2 , и выполняется неравенство

y< x1 + x2 < sup X1 + sup X 2 . Это означает, что существует элемент

x = +x1 x2 ∈ X1+ X2 which is greater than y and

sup X1 + sup X 2 = sup (X1 + X 2) .◄

The proofs of the remaining properties are carried out in a similar way and

lie to the reader.

§ 6 Countable and uncountable sets

Definition 1.6.1. Consider the set of the first n natural numbers

n = (1,2,..., n) and some set A . If it is possible to establish mutually

one-to-one correspondence between A and n , then the set A will be called

final.

Definition 1.6.2. Let some set A be given. If I may

establish a one-to-one correspondence between the set A and

set of natural numbers, then the set A will be called a count

Definition 1.6.3. If the set A is finite or countable, then we will

say that it is nothing more than countable.

Thus, a set will be countable if its elements can be counted.

put in sequence.

Example 1. The set of even numbers is countable, since the mapping n ↔ 2n

is a one-to-one correspondence between the set of natural

numbers and a set of even numbers.

Obviously, such a correspondence can be established not in the only way

zom. For example, you can establish a correspondence between a set and a set

(integer numbers), establishing a correspondence in this way

Axiom of continuity (completeness). $A \subset \mathbb(R)$ and $B \subset \mathbb(R)$ $a\in A$ and $b \in B$ the inequality $a \leqslant b$ , there is a real number $\xi$ that for everyone $a\in A$ and $b \in B$ there is a relation

$a \leqslant \xi \leqslant b$

Geometrically, if we treat real numbers as points on a line, this statement seems obvious. If two sets $A$ and $B$ are such that on the number line all elements of one of them lie to the left of all elements of the second, then there is a number $\xi$ , separating these two sets, that is, lying to the right of all elements $A$ (except perhaps the $\xi$ ) and to the left of all elements $B$ (same clause).

It should be noted here that despite the "obviousness" of this property, for rational numbers it is not always satisfied. For example, consider two sets:

A = $x \in \mathbb(Q): x > 0, \; x^2< 2\}, \quad B = \{x \in \mathbb{Q}: x >0,\; x^2 > 2$

It is easy to see that for any elements $a\in A$ and $b \in B$ the inequality $a< b$ . However rational numbers $\xi$ , separating these two sets, does not exist. Indeed, this number can only be $\sqrt(2)$ , but it is not rational .

The role of the axiom of continuity in the construction of mathematical analysis

(Weierstrass theorem). Every bounded monotonically increasing sequence converges
(Bolzano-Cauchy theorem). A function continuous on a segment that takes on values of different signs at its ends vanishes at some interior point of the segment
(Existence of power, exponential, logarithmic and all trigonometric functions over the entire "natural" domain of definition). For example, it is proved that for every $a > 0$ and whole $n \geqslant 1$ exist $\sqrt[n](a)$ , that is, the solution of the equation $x^n=a, x>0$ . This allows you to determine the value of the expression $a^x$ for all rational $x$ :

$a^(m/n) = \left(\sqrt[n](a)\right)^m$

Finally, again due to the continuity of the number line, one can determine the value of the expression $a^x$ already for arbitrary $x \in \R$ . Similarly, using the continuity property, we prove the existence of the number $\log_(a)(b)$ for any $a,b >0 , a\neq 1$ .

For a long historical period of time, mathematicians proved theorems from analysis, in “thin places” referring to the geometric justification, and more often skipping them altogether, since it was obvious. The essential concept of continuity was used without any clear definition. Only in the last third of the 19th century did the German mathematician Karl Weierstrass produce the arithmetization of analysis, constructing the first rigorous theory of real numbers as infinite decimal fractions. He proposed the classical definition of the limit in the language $\varepsilon - \delta$ , proved a number of statements that were considered “obvious” before him, and thus completed the construction of the foundation of mathematical analysis.

Later, other approaches to the definition of a real number were proposed. In the axiomatic approach, the continuity of real numbers is explicitly singled out as a separate axiom. In constructive approaches to real number theory, such as when constructing real numbers using Dedekind sections, the continuity property (in one formulation or another) is proved as a theorem.

Other Statements of the Continuity Property and Equivalent Propositions

Continuity according to Dedekind

The question of the continuity of real numbers Dedekind considers in his work " Continuity and irrational numbers". In it he compares the rational numbers with the points of a straight line. As you know, between rational numbers and points of a straight line, you can establish a correspondence when the starting point and unit of measurement of the segments are chosen on the straight line. With the help of the latter, for every rational number $a$ construct the corresponding segment, and putting it aside to the right or to the left, depending on whether there is $a$ positive or negative number, get point $p$ corresponding to the number $a$ . So every rational number $a$ matches one and only one point $p$ on a straight line.

To find out what this continuity consists of, Dedekind makes the following remark. If a $p$ is a certain point of the line, then all points of the line fall into two classes: points located to the left $p$ , and points to the right $p$ . The very point $p$ can be arbitrarily assigned to either the lower or the upper class. Dedekind sees the essence of continuity in the reverse principle:

Finite cover lemma (Heine-Borel principle)

Finite Cover Lemma (Heine - Borel). In any system of intervals covering a segment, there is a finite subsystem covering this segment.

Limit point lemma (Bolzano-Weierstrass principle)

Limit Point Lemma (Bolzano - Weierstrass). Every infinite bounded number set has at least one limit point.

Equivalence of sentences expressing the continuity of the set of real numbers

Let's make some preliminary remarks. According to the axiomatic definition of a real number, the collection of real numbers satisfies three groups of axioms. The first group is the field axioms. The second group expresses the fact that the set of real numbers is a linearly ordered set, and the order relation is consistent with the basic operations of the field. Thus, the first and second groups of axioms mean that the set of real numbers is an ordered field. The third group of axioms consists of one axiom - the axiom of continuity (or completeness).

To show the equivalence of different formulations of the continuity of the real numbers, one must prove that if one of these propositions holds for an ordered field, then all the others are true.

Theorem. Let be $\mathsf(R)$ - an arbitrary linearly ordered set . The following statements are equivalent:

Whatever the non-empty sets $A \subset \mathsf(R)$ and $B \subset \mathsf(R)$ , such that for any two elements $a\in A$ and $b \in B$ the inequality $a \leqslant b$ , there is such an element $\xi \in \mathsf(R)$ that for everyone $a\in A$ and $b \in B$ there is a relation $a \leqslant \xi \leqslant b$
For any section in $\mathsf(R)$ there is an element that produces this section
Every non-empty set bounded above $A \subset \mathsf(R)$ has a supremum
Every non-empty set bounded below $A \subset \mathsf(R)$ has an infimum

As can be seen from this theorem, these four sentences use only what is on $\mathsf(R)$ introduced a linear order relation, and do not use the field structure. Thus, each of them expresses the property $\mathsf(R)$ as a linearly ordered set. This property (of an arbitrary linearly ordered set, not necessarily the set of real numbers) is called continuity, or completeness, according to Dedekind.

Proving the equivalence of other sentences already requires a field structure.

Theorem. Let be $\mathsf(R)$ - an arbitrary ordered field. The following sentences are equivalent:

$\mathsf(R)$ (as a linearly ordered set) is Dedekind complete
For $\mathsf(R)$ fulfilled the principle of Archimedes and principle of nested segments
For $\mathsf(R)$ the Heine-Borel principle is fulfilled
For $\mathsf(R)$ the Bolzano-Weierstrass principle is fulfilled

The proof of the above theorems can be found in the books from the bibliography given below.

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Notes

Literature

Kudryavtsev, L. D. Course of mathematical analysis. - 5th ed. - M .: "Drofa", 2003. - T. 1. - 704 p. - ISBN 5-7107-4119-1.
Fikhtengolts, G. M. Fundamentals of mathematical analysis. - 7th ed. - M .: "FIZMATLIT", 2002. - T. 1. - 416 p. - ISBN 5-9221-0196-X.
Dedekind, R.= Stetigkeit und irrationale Zahlen. - 4th revised edition. - Odessa: Mathesis, 1923. - 44 p.
Zorich, V. A. Mathematical analysis. Part I. - Ed. 4th, corrected .. - M .: "MTsNMO", 2002. - 657 p. - ISBN 5-94057-056-9.
Continuity of functions and numerical domains: B. Bolzano, L. O. Cauchy, R. Dedekind, G. Kantor. - 3rd ed. - Novosibirsk: ANT, 2005. - 64 p.

An excerpt characterizing the Continuity of the set of real numbers

- So that's who I feel sorry for - human dignity, peace of mind, purity, and not their backs and foreheads, which, no matter how much you flog, no matter how you shave, everything will remain the same backs and foreheads.
“No, no, and a thousand times no, I will never agree with you,” said Pierre.

In the evening, Prince Andrei and Pierre got into a carriage and drove to the Bald Mountains. Prince Andrei, looking at Pierre, occasionally interrupted the silence with speeches that proved that he was in a good mood.
He told him, pointing to the fields, about his economic improvements.
Pierre was gloomy silent, answering in monosyllables, and seemed immersed in his own thoughts.
Pierre thought that Prince Andrei was unhappy, that he was mistaken, that he did not know the true light, and that Pierre should come to his aid, enlighten and raise him. But as soon as Pierre figured out how and what he would say, he had a premonition that Prince Andrei would drop everything in his teachings with one word, with one argument, and he was afraid to start, he was afraid to expose his beloved shrine to the possibility of ridicule.
“No, why do you think,” Pierre suddenly began, lowering his head and taking the form of a butting bull, why do you think so? You shouldn't think like that.
– What am I thinking about? Prince Andrew asked with surprise.
- About life, about the purpose of a person. It can't be. That's what I thought, and it saved me, you know what? freemasonry. No, you don't smile. Freemasonry is not a religious, not a ritual sect, as I thought, but Freemasonry is the best, the only expression of the best, eternal aspects of humanity. - And he began to explain to Prince Andrei Freemasonry, as he understood it.
He said that Freemasonry is the teaching of Christianity, freed from state and religious shackles; the doctrine of equality, brotherhood and love.
– Only our holy brotherhood has a real meaning in life; everything else is a dream,” said Pierre. - You understand, my friend, that outside this union everything is full of lies and untruths, and I agree with you that there is nothing left for a smart and kind person, as soon as, like you, to live out his life, trying only not to interfere with others. But assimilate our basic convictions, join our brotherhood, give yourself to us, let yourself be led, and now you will feel, as I felt, a part of this huge, invisible chain, of which the beginning is hidden in heaven, - said Pierre.
Prince Andrei, silently, looking in front of him, listened to Pierre's speech. Several times, not hearing the noise of the carriage, he asked Pierre for unheard words. From the special brilliance that lit up in the eyes of Prince Andrei, and from his silence, Pierre saw that his words were not in vain, that Prince Andrei would not interrupt him and would not laugh at his words.
They drove up to a flooded river, which they had to cross by ferry. While the carriage and horses were being installed, they went to the ferry.
Prince Andrei, leaning on the railing, silently looked along the flood shining from the setting sun.
- Well, what do you think about it? - asked Pierre, - why are you silent?
- What I think? I listened to you. All this is so, - said Prince Andrei. - But you say: join our brotherhood, and we will show you the purpose of life and the purpose of man, and the laws that govern the world. But who are we people? Why do you know everything? Why am I the only one who doesn't see what you see? You see the kingdom of goodness and truth on earth, but I do not see it.
Pierre interrupted him. Do you believe in a future life? - he asked.
- To the next life? - repeated Prince Andrei, but Pierre did not give him time to answer and took this repetition for a denial, especially since he knew the former atheistic convictions of Prince Andrei.
– You say that you cannot see the realm of goodness and truth on earth. And I did not see him, and you cannot see him if you look at our life as the end of everything. On earth, precisely on this earth (Pierre pointed to the field), there is no truth - everything is a lie and evil; but in the world, in the whole world, there is a kingdom of truth, and we are now the children of the earth, and forever the children of the whole world. Do I not feel in my soul that I am part of this vast, harmonious whole. Do I not feel that I am in this vast, innumerable number of beings in which the Divine is manifested - the highest power, as you like - that I am one link, one step from lower beings to higher ones. If I see, I clearly see this ladder that leads from the plant to man, then why should I suppose that this ladder is interrupted with me, and does not lead further and further. I feel that not only can I not disappear, just as nothing in the world disappears, but that I will always be and have always been. I feel that besides me, spirits live above me and that there is truth in this world.
“Yes, this is the teaching of Herder,” said Prince Andrei, “but not that, my soul, will convince me, but life and death, that’s what convinces. It convinces that you see a creature dear to you, who is connected with you, before whom you were guilty and hoped to justify yourself (Prince Andrei trembled in his voice and turned away) and suddenly this creature suffers, suffers and ceases to be ... Why? It cannot be that there is no answer! And I believe he is... That's what convinces, that's what convinced me, - said Prince Andrei.
“Well, yes, yes,” said Pierre, “isn’t that what I say too!”
- Not. I only say that it is not arguments that convince you of the need for a future life, but when you walk in life hand in hand with a person, and suddenly this person disappears into nowhere, and you yourself stop in front of this abyss and look into it. And I looked...
- Well, so what! Do you know what is there and what is someone? There is a future life. Someone is God.
Prince Andrew did not answer. The carriage and horses had long since been brought to the other side and had already been laid down, and the sun had already disappeared to half, and the evening frost covered the puddles near the ferry with stars, and Pierre and Andrei, to the surprise of the lackeys, coachmen and carriers, were still standing on the ferry and talking.
- If there is a God and there is a future life, then there is truth, there is virtue; and the highest happiness of man is to strive to achieve them. We must live, we must love, we must believe, - said Pierre, - that we do not live now only on this piece of land, but we have lived and will live forever there in everything (he pointed to the sky). Prince Andrei stood leaning on the railing of the ferry and, listening to Pierre, without taking his eyes off, looked at the red reflection of the sun over the blue flood. Pierre is silent. It was completely quiet. The ferry had landed long ago, and only the waves of the current with a faint sound hit the bottom of the ferry. It seemed to Prince Andrei that this rinsing of the waves was saying to Pierre's words: "True, believe this."
Prince Andrei sighed, and with a radiant, childish, tender look looked into Pierre's flushed, enthusiastic, but still timid in front of his superior friend.
“Yes, if that were the case!” - he said. “However, let’s go sit down,” Prince Andrei added, and leaving the ferry, he looked at the sky, which Pierre pointed out to him, and for the first time, after Austerlitz, he saw that high, eternal sky, which he saw lying on the Austerlitz field, and something long asleep, something the best that was in him, suddenly awoke joyfully and youthfully in his soul. This feeling disappeared as soon as Prince Andrei entered the habitual conditions of life again, but he knew that this feeling, which he did not know how to develop, lived in him. A meeting with Pierre was for Prince Andrei an epoch from which, although in appearance it was the same, but in the inner world, his new life began.

It was already getting dark when Prince Andrei and Pierre drove up to the main entrance of the Lysogorsky house. While they were driving up, Prince Andrei with a smile drew Pierre's attention to the turmoil that had taken place at the back porch. A bent old woman with a knapsack on her back, and a short man in a black robe and with long hair, seeing a carriage driving in, rushed to run back through the gate. Two women ran after them, and all four, looking back at the carriage, ran frightened up the back porch.
“These are God’s Machines,” said Prince Andrei. They took us for their father. And this is the only thing in which she does not obey him: he orders to drive these wanderers, and she accepts them.
- What are God's people? Pierre asked.
Prince Andrei did not have time to answer him. The servants went out to meet him, and he asked where the old prince was and how soon they were waiting for him.
The old prince was still in the city, and they were waiting for him every minute.
Prince Andrei led Pierre to his quarters, which always awaited him in perfect order in his father's house, and he himself went to the nursery.
“Let's go to my sister,” said Prince Andrei, returning to Pierre; - I have not seen her yet, she is now hiding and sitting with her God people. Serve her right, she will be embarrassed, and you will see God's people. C "est curieux, ma parole. [This is curious, honestly.]
- Qu "est ce que c" est que [What is] God's people? Pierre asked.
- But you'll see.
Princess Mary was really embarrassed and blushed in spots when they entered her. In her comfortable room with lamps in front of the icon cases, on the sofa, at the samovar, sat next to her a young boy with a long nose and long hair, and in a monastic cassock.
On an armchair, beside him, sat a wrinkled, thin old woman with a meek expression of a child's face.
- Andre, pourquoi ne pas m "avoir prevenu? [Andrey, why didn’t they warn me?] - she said with meek reproach, standing in front of her wanderers, like a hen in front of chickens.
– Charmee de vous voir. Je suis tres contente de vous voir, [Very glad to see you. I am so pleased to see you,] she said to Pierre, while he was kissing her hand. She knew him as a child, and now his friendship with Andrei, his misfortune with his wife, and most importantly, his kind, simple face, endeared her to him. She looked at him with her beautiful, radiant eyes and seemed to say: "I love you very much, but please don't laugh at mine." After exchanging the first phrases of greeting, they sat down.
“Ah, and Ivanushka is here,” said Prince Andrei, pointing with a smile at the young wanderer.
– Andrew! said Princess Mary pleadingly.
- Il faut que vous sachiez que c "est une femme, [Know that this is a woman] - said Andrei to Pierre.
Andre, au nom de Dieu! [Andrey, for God's sake!] - repeated Princess Marya.
It was evident that Prince Andrei's mocking attitude towards the wanderers and Princess Mary's useless intercession for them were habitual, established relations between them.
- Mais, ma bonne amie, - said Prince Andrei, - vous devriez au contraire m "etre reconaissante de ce que j" explique a Pierre votre intimite avec ce jeune homme ... [But, my friend, you should be grateful to me that I explain to Pierre your closeness to this young man.]
– Vrayment? [Really?] - Pierre said curiously and seriously (for which Princess Mary was especially grateful to him), peering through glasses at Ivanushka's face, who, realizing that it was about him, looked around at everyone with cunning eyes.
Princess Marya was quite unnecessarily embarrassed for her own people. They didn't hesitate at all. The old woman, lowering her eyes, but glancing askance at the newcomers, knocking her cup upside down on a saucer and placing a bitten piece of sugar beside her, calmly and motionlessly sat on her chair, waiting to be offered more tea. Ivanushka, drinking from a saucer, looked at the young people with sly, feminine eyes from under his brows.
- Where, in Kyiv was? Prince Andrei asked the old woman.
- There was, father, - the old woman answered loquaciously, - on Christmas itself, she was honored with the saints, heavenly secrets from the saints. And now from Kolyazin, father, great grace has opened ...
- Well, is Ivanushka with you?
“I’m walking on my own, breadwinner,” Ivanushka said, trying to speak in a bass voice. - Only in Yukhnov did they agree with Pelageyushka ...
Pelageyushka interrupted her comrade; She seemed to want to tell what she saw.
- In Kolyazin, father, great grace has opened.
- Well, new relics? asked Prince Andrew.
“Enough, Andrei,” said Princess Mary. - Don't tell me, Pelageushka.
- No ... what are you, mother, why not tell? I love him. He is kind, exacted by God, he gave me, a benefactor, rubles, I remember. As I was in Kyiv, Kiryusha the holy fool tells me - truly a man of God, he walks barefoot in winter and summer. Why are you walking, he says, out of your place, go to Kolyazin, there is a miraculous icon, Mother Blessed Virgin Mary has opened. With those words, I said goodbye to the saints and went ...
Everyone was silent, one wanderer spoke in a measured voice, drawing in air.
- My father, the people came to me and they say: great grace has opened, at Mother Blessed Virgin Mary drops from her cheek ...
“Well, well, well, you’ll tell me later,” Princess Marya said, blushing.
“Let me ask her,” said Pierre. - Did you see it yourself? - he asked.
- How, father, she herself was honored. The radiance on her face is like the light of heaven, and from mother’s cheek it drips and drips ...
“But this is a deception,” Pierre said naively, listening attentively to the wanderer.
“Ah, father, what are you talking about!” - Pelageyushka said with horror, turning to Princess Marya for protection.
“They are deceiving the people,” he repeated.
- Lord Jesus Christ! – crossed said the stranger. “Oh, don’t talk, father. So one anaral did not believe, said: “the monks are deceiving”, but as he said, he went blind. And he dreamed that Mother Pecherskaya came to him and said: "Trust me, I will heal you." So he began to ask: take me and take me to her. I'm telling you the truth, I saw it myself. They brought him blind right to her, came up, fell down, said: “heal! I will give it to you, he says, in what the king complained. I saw it myself, father, the star is embedded in it like that. Well, it has dawned! It's wrong to say that. God will punish, ”she addressed Pierre instructively.
- How did the star find itself in the image? Pierre asked.
- Did you make your mother a general? - said Prince Andrei smiling.
Pelageushka suddenly turned pale and clasped her hands.
“Father, father, sin on you, you have a son!” she spoke, suddenly turning from pallor into a bright color.
- Father, what did you say, God forgive you. - She crossed herself. “God, forgive him. Mother, what is this? ... - she turned to Princess Marya. She got up and almost crying began to collect her purse. She was evidently both frightened and ashamed that she enjoyed the blessings in the house where they could say this, and it was a pity that she now had to be deprived of the blessings of this house.
- Well, what are you looking for? - said Princess Mary. Why did you come to me?...
“No, I’m joking, Pelageushka,” said Pierre. - Princesse, ma parole, je n "ai pas voulu l" offerr, [Princess, I really didn’t want to offend her,] I just did. Don't think, I was joking, - he said, smiling timidly and wanting to make amends for his guilt. - After all, it's me, and he was just joking.
Pelageyushka stopped incredulously, but there was such sincerity of repentance in Pierre's face, and Prince Andrei looked so meekly at Pelageyushka and then at Pierre that she gradually calmed down.

The wanderer calmed down and, brought back to conversation, then talked for a long time about Father Amphilochius, who was such a holy life that his hand smelled of his hand, and how the monks she knew on her last journey to Kyiv gave her the keys to the caves, and how she, taking crackers with her, spent two days in caves with saints. “I will pray to one, I will read, I will go to another. Pine, I’ll go and kiss again; and such, mother, silence, such grace that you don’t even want to go out into the light of God.
Pierre listened to her attentively and seriously. Prince Andrei left the room. And after him, leaving the people of God to finish their tea, Princess Mary led Pierre into the living room.
“You are very kind,” she told him.
“Ah, I really didn’t think to offend her, as I understand and highly appreciate these feelings!
Princess Mary looked at him silently and smiled tenderly. “After all, I have known you for a long time and love you like a brother,” she said. How did you find Andrew? she asked hastily, not giving him time to say anything in response to her kind words. “He worries me a lot. His health is better in winter, but last spring the wound opened, and the doctor said that he must go for treatment. And morally, I'm very afraid for him. He is not a character like us women to suffer and cry out his grief. He carries it inside himself. Today he is cheerful and lively; but it was your arrival that had such an effect on him: he is rarely like that. If you could persuade him to go abroad! He needs activity, and this smooth, quiet life is ruining him. Others do not notice, but I see.
At 10 o'clock the waiters rushed to the porch, hearing the bells of the old prince's carriage approaching. Prince Andrei and Pierre also went out onto the porch.
- Who is this? asked the old prince, getting out of the carriage and guessing Pierre.
– AI is very happy! kiss, - he said, having learned who the unfamiliar young man was.
The old prince was in a good spirit and kindly treated Pierre.
Before dinner, Prince Andrei, returning back to his father's study, found the old prince in a heated argument with Pierre.
Pierre argued that the time would come when there would be no more war. The old prince, teasing, but not angry, challenged him.
- Let the blood out of the veins, pour water, then there will be no war. Woman’s nonsense, woman’s nonsense, ”he said, but still affectionately patted Pierre on the shoulder, and went up to the table, at which Prince Andrei, apparently not wanting to enter into a conversation, was sorting through the papers brought by the prince from the city. The old prince approached him and began to talk about business.
- The leader, Count Rostov, did not deliver half of the people. He came to the city, decided to call for dinner, - I asked him such a dinner ... But look at this one ... Well, brother, - Prince Nikolai Andreevich turned to his son, clapping Pierre on the shoulder, - well done your friend, I fell in love with him! Fires me up. The other one speaks smart words, but I don’t want to listen, but he lies and inflames me, old man. Well, go, go, - he said, - maybe I will come, I will sit at your supper. I'll bet again. Love my fool, Princess Mary, ”he shouted to Pierre from the door.
Pierre now only, on his visit to the Bald Mountains, appreciated the full strength and charm of his friendship with Prince Andrei. This charm was expressed not so much in his relations with himself, but in relations with all relatives and household. Pierre, with the old, stern prince and with the meek and timid Princess Mary, despite the fact that he hardly knew them, immediately felt like an old friend. They all already loved him. Not only Princess Mary, bribed by his meek attitude towards wanderers, looked at him with the most radiant eyes; but the little, one-year-old Prince Nikolai, as his grandfather called him, smiled at Pierre and went into his arms. Mikhail Ivanovich, m lle Bourienne looked at him with joyful smiles when he talked with the old prince.