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. . ,xn , . . . . Take the point x1 and on the interval [0, 1] = I0 fix a closed interval ofpositive length I1 not containing the point x1 . In the interval I1 construct an intervalI2 not containing x2 . If the interval In has been constructed, then, since |In | > 0, we/ In+1 and |In+1 | > 0. By the nested setconstruct in it an interval In+1 so that xn+1 ∈lemma, there is a point c belonging to all of the intervals I0 , I1 , . . . , In , . . . . But thispoint of the closed interval I0 = [0, 1] by construction cannot be any point of thesequence x1 , x2 , . .
. , xn , . . . .Corollaries1) Q = R, and so irrational numbers exist.2) There exist transcendental numbers, since the set of algebraic numbers iscountable.(After solving Exercise 3 below, the reader will no doubt wish to reinterpret thislast proposition, stating it as follows: Algebraic numbers are occasionally encountered among the real numbers.)At the very dawn of set theory the question arose whether there exist sets ofcardinality between countable sets and sets having cardinality of the continuum,and the conjecture was made, known as the continuum hypothesis, that there are nointermediate cardinalities.13 Fromthe Latin continuum, meaning continuous, or solid.2.4 Countable and Uncountable Sets77The question turned out to involve the deepest parts of the foundations of mathematics.
It was definitively answered in 1963 by the American mathematician P. Cohen. Cohen proved that the continuum hypothesis is undecidable by showing thatneither the hypothesis nor its negation contradicts the standard axiom system of settheory, so that the continuum hypothesis can be neither proved nor disproved withinthat axiom system. This situation is very similar to the way in which Euclid’s fifthpostulate on parallel lines is independent of the other axioms of geometry.2.4.3 Problems and Exercises1.
Show that the set of real numbers has the same cardinality as the points of theinterval ]−1, 1[.2. Give an explicit one-to-one correspondence betweena)b)c)d)the points of two open intervals;the points of two closed intervals;the points of a closed interval and the points of an open interval;the points of the closed interval [0, 1] and the set R.3. Show thata) every infinite set contains a countable subset;b) the set of even integers has the same cardinality as the set of all natural numbers;c) the union of an infinite set and an at most countable set has the same cardinality as the original infinite set;d) the set of irrational numbers has the cardinality of the continuum;e) the set of transcendental numbers has the cardinality of the continuum.4.
Show thata) the set of increasing sequences of natural numbers {n1 < n2 < · · · } has thesame cardinality as the set of fractions of the form 0.α1 α2 . . .;b) the set of all subsets of a countable set has cardinality of the continuum.5. Show thata) the set P(X) of subsets of a set X has the same cardinality as the set of allfunctions on X with values 0, 1, that is, the set of mappings f : X → {0, 1};b) for a finite set X of n elements, card P(X) = 2n ;c) taking account of the results of Exercises 4b) and 5a), one can writecard P(X) = 2card X , and, in particular, card P(N) = 2card N = card R;d) for any set Xcard X < 2card X ,in particular,Hint: See Cantor’s theorem in Sect.
1.4.1.n < 2nfor any n ∈ N.782The Real Numbers6. Let X1 , . . . , Xn be a finite system of finite sets. Show thatm Xi =card Xi1 −card(Xi1 ∩ Xi2 ) +cardi=1i1+i1 <i2card(Xi1 ∩ Xi2 ∩ Xi3 ) − · · · +i1 <i2 <i3+ (−1)m−1 card(X1 ∩ · · · ∩ Xm ),the summation extending over all sets of indices from 1 to m satisfying the inequalities under the summation signs.7. On the closed interval [0, 1] ⊂ R describe the sets of numbers x ∈ [0, 1] whoseternary representation x = 0.α1 α2 α3 .
. . , αi ∈ {0, 1, 2}, has the property:a) α1 = 1;b) (α1 = 1) ∧ (α2 = 1);c) ∀i ∈ N (αi = 1) (the Cantor set).8. (Continuation of Exercise 7.) Show thata) the set of numbers x ∈ [0, 1] whose ternary representation does not contain 1has the same cardinality as the set of all numbers whose binary representation hasthe form 0.β1 β2 . . .;b) the Cantor set has the same cardinality as the closed interval [0, 1].Chapter 3LimitsIn discussing the various aspects of the concept of a real number we remarked inparticular that in measuring real physical quantities we obtain sequences of approximate values with which one must then work.Such a state of affairs immediately raises at least the following three questions:1) What relation does the sequence of approximations so obtained have to the quantity being measured? We have in mind the mathematical aspect of the question,that is, we wish to obtain an exact expression of what is meant in general bythe expression “sequence of approximate values” and the extent to which such asequence describes the value of the quantity.
Is the description unambiguous, orcan the same sequence correspond to different values of the measured quantity?2) How are operations on the approximate values connected with the same operations on the exact values, and how can we characterize the operations that canlegitimately be carried out by replacing the exact values with approximate ones?3) How can one determine from a sequence of numbers whether it can be a sequence of arbitrarily precise approximations of the values of some quantity?The answer to these and related questions is provided by the concept of the limitof a function, one of the fundamental concepts of analysis.We begin our discussion of the theory of limits by considering the limit of afunction of a natural-number argument (a sequence), in view of the fundamentalrole played by these functions, as already explained, and also because all the basicfacts of the theory of limits can actually be clearly seen in this simplest situation.3.1 The Limit of a Sequence3.1.1 Definitions and ExamplesWe recall the following definition.Definition 1 A function f : N → X whose domain of definition is the set of naturalnumbers is called a sequence.© Springer-Verlag Berlin Heidelberg 2015V.A.
Zorich, Mathematical Analysis I, Universitext,DOI 10.1007/978-3-662-48792-1_379803LimitsThe values f (n) of the function f are called the terms of the sequence. It is customary to denote them by a symbol for an element of the set into which the mappinggoes, endowing each symbol with the corresponding index of the argument. Thus,xn := f (n). In this connection the sequence itself is denoted {xn }, and also writtenas x1 , x2 , .
. . , xn , . . . . It is called a sequence in X or a sequence of elements of X.The element xn is called the nth term of the sequence.Throughout the next few sections we shall be considering only sequences f :N → R of real numbers.Definition 2 A number A ∈ R is called the limit of the numerical sequence {xn } iffor every neighborhood V (A) of A there exists an index N (depending on V (A))such that all terms of the sequence having index larger than N belong to the neighborhood V (A).We shall give an expression in formal logic for this definition below, but we firstpoint out another common formulation of the definition of the limit of a sequence.A number A ∈ R is called the limit of the sequence {xn } if for every ε > 0 thereexists an index N such that |xn − A| < ε for all n > N .The equivalence of these two statements is easy to verify (verify it!) if we remarkthat any neighborhood V (A) of A contains some ε-neighborhood of the point A.The second formulation of the definition of a limit means that no matter whatprecision ε > 0 we have prescribed, there exists an index N such that the absoluteerror in approximating the number A by terms of the sequence {xn } is less than ε assoon as n > N .We now write these formulations of the definition of a limit in the language ofsymbolic logic, agreeing that the expression “limn→∞ xn = A” is to mean that A isthe limit of the sequence {xn }.
Thuslim xn = A := ∀V (A) ∃N ∈ N ∀n > N xn ∈ V (A)n→∞and respectivelylim xn = A := ∀ε > 0 ∃N ∈ N ∀n > N |xn − A| < ε .n→∞Definition 3 If limn→∞ xn = A, we say that the sequence {xn } converges to A ortends to A and write xn → A as n → ∞.A sequence having a limit is said to be convergent. A sequence that does not havea limit is said to be divergent.Let us consider some examples.Example 1 limn→∞1 We1n= 0, since | n1 − 0| =1n< ε when n > N = [ 1ε ].1recall that [x] is the integer part of the number x. (See Corollaries 70 and 100 of Sect. 2.2.)3.1 The Limit of a SequenceExample 2 limn→∞n+1n81= 1, since | n+1n − 1| =1nn< ε if n > [ 1ε ].n(−1)Example 3 limn→∞ (1 + (−1)n ) = 1, since |(1 + n ) − 1| =Example 4 limn→∞sin nnExample 5 limn→∞1qn= 0, since | sinn n − 0| ≤1n1n< ε when n > [ 1ε ].< ε for n > [ 1ε ].= 0 if |q| > 1.Let us verify this last assertion using the definition of the limit.
As was shown inParagraph c of Sect. 2.2.4, for every ε > 0 there exists N ∈ N such that |q|1N < ε.Since |q| > 1, we shall have | q1n − 0| < |q|1 n <in the definition of the limit is satisfied.1|q|N< ε for n > N , and the conditionnExample 6 The sequence 1, 2, 13 , 4, 15 , 6, 17 , . . . whose nth term is xn = n(−1) ,n ∈ N, is divergent.Proof Indeed, if A were the limit of this sequence, then, as follows from the definition of limit, any neighborhood of A would contain all but a finite number of termsof the sequence.A number A = 0 cannot be the limit of this sequence; for if ε = |A|2 > 0, all11the terms of the sequence of the form 2k+1for which 2k+1< |A|lieoutsidethe2ε-neighborhood of A.But the number 0 also cannot be the limit, since, for example, there are infinitelymany terms of the sequence lying outside the 1-neighborhood of 0.Example 7 One can verify similarly that the sequence 1, −1, +1, −1, .
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