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MAT425 Real Analysis L16

Prof. Thistleton



Recall the Bolzano-Weierstrass Theorem:











We may restate this theorem in the context of sequences. We will show that every bounded sequence has a sequential cluster point. First recall what we mean when we say that c is a cluster point of the set S.








Recall also our previous lemma:


The point c is a cluster point of the set S if and only if there is a sequence of elements of S, all different from c, and converging to c.


We modify this lemma and present a result in terms of subsequences. Denote by \( \{x_n\} \) the range of the sequence (xn).

Theorem 201

Let c be a cluster point of \( \{x_n\} \).Then there is a subsequence (xnk) which converges to c.


Definition 206

A point c is termed a sequential cluster point of the sequence (xn) if \( \{ n: x_n \in (c-\epsilon,c+\epsilon)\} \)if infinite for each \( \epsilon \gt 0 \).

Theorem 211

c is a sequential cluster point of (xn) if and only if (xn) has a subsequence which converges to c .


























Bolzano-Weierstrass Theorem for Sequences

Theorem 212

Every bounded sequence has a sequential cluster point.








Theorem 213

Every bounded sequence has a convergent subsequence.














Cauchy Sequences We have seen that, if lim xn = L, then, given \( \epsilon \gt 0 \)there is a natural number N such that, \( n,m \gt N \Rightarrow \vert x_n - x_m\vert < \epsilon\).We use this idea to define a Cauchy sequence.

Definition 222

A sequence (xn) is called a Cauchy sequence if, for any \( \epsilon \gt 0 \), there exists a natural number N so that \( n,m \gt N \Rightarrow \vert x_n - x_m\vert < \epsilon\).

Our previous result may then be stated as follows: A convergent sequence is a Cauchy sequence.


This raises the obvious question: Is a Cauchy sequence a convergent sequence? We answer this question bit by bit.

Lemma 226

Let (xn) be a Cauchy sequence. If (xn) has a convergent subsequence then (xn) itself converges.


Lemma 229

Let (xn) be a Cauchy sequence. Then (xn) is bounded.


























Theorem 231

A Cauchy sequence of real numbers converges in \( \Re \).


We now begin to use our knowledge of sequences and topology to understand some of the most basic results in Calculus. Two major results will be the Extreme value Theorem (do you remember what this says?) and the Heine-Borel Theorem.


We begin by defining what we mean by a compact set.

Definition 234

A set \( K \subseteq \Re\) is said to be compact if every continuous function \( f:K\rightarrow\Re \) assumes a maximum.




















Theorem 238

Let A and B be compact sets. Then \(A \cup B\) is compact.




















Theorem 239

A compact set is closed and bounded.


 
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William Thistleton
11/10/1998