Analysis 2 (Springer-Lehrbuch) by Stefan Hildebrandt

By Stefan Hildebrandt

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Similarly, if x > b there exists N 0 such that bN 0 < x. Thus x … ŒaN 0 ; bN 00 . Therefore Œa; b D \1 nD Œan ; bn . Definition 2. A point x is a cluster point (or accumulation point) of the sequence xj < ": fxn g1 nD1 if given any " > 0 there are infinitely many indices n such that jxn Thus, given any open interval J centered at x and any integer n there exists m > n such that xm 2 J. We can characterize a cluster point of fxn g1 nD1 in terms of its convergent subsequences: Proposition 2. 1 nk nD1 Proof.

Assume that L D sup S. 1 xn D L. The statement about the greatest lower bound is justified in a similar manner. The least upper bound of a set need not belong to that set. For example, if SD 1 1 W n D 1; 2; 3; : : : ; n then sup S D 1, but 1 … S. If the least upper bound of a set S belongs to S, we will say that sup S is the maximum value of the numbers in S and may use the notation max S. Similarly, if the greatest lower bound of a set S belongs to S, we will say that inf S is the minimum value of the numbers in S and may use the notation min S.

4 The Cauchy Convergence Criterion 41 and let xn be a rational number such that  max an ; xn 1 ; yn 1 n à < xn < yn : In all cases xn Ä xn < yn < yn 1 ; 1 xn > an or yn < an ; yn xn < 1 : n 1 This completes the inductive construction of the sequences fxn g1 nD1 and fyn gnD1 with the properties that lead to the conclusions in the statement of Theorem 2. The above proof can be found in the book by Bishop, Bridges and Douglas (Constructive Analysis, published by Springer). The book contains the elegant construction of real numbers as Cauchy sequences of rational numbers.

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