Question

Use the Heine-Borel Theorem to prove the Bolzano-Weierstrass The- orem.

Answer 1

ANSWER :

Proof :  

Proof of Bolzano - Weierstrass Theorem from Heine - Borel Theorem,

Let
{an
be a bounded sequences IR. Then $a_{n} \epsilon \left [ c,d \right ]$ for some $c,d \epsilon IR \, \, \, c,d$ To prove applying contradiction .

Assume
{an
contains no convergend subsequence
in c, d
, then $\forall x\, \epsilon \left [ c.d \right ],\exists\, \, \epsilon > 0$ such that  $B_{\epsilon }(x)=\left ( x-\epsilon ,x\rightarrow \epsilon \right )$

Contains finitely many $a_{n}$ .

Otherwise if there excused an x such that $\left | x-a_{n} \right |< \epsilon\, \, \, \forall \rightarrow 0$ and finitely many $a_{n}$ .

$\Rightarrow$$a_{n}$ would have a convergent subsequence contaning.

Then,

$\Rightarrow \bigcup _{x\epsilon [c,d]}B_{\epsilon }(x)\supset [c,d]$

Then by HBT a finite subcover such that ,

$\bigcup_{i=1}^h{}B_{\epsilon }(xi)\supset [c,d]$

Therefore, is contained subcover.

{an

There we conclude that

has finitely marry.

{an

$Memvers \, \, bounded \Rightarrow \Leftarrow$

Here, $\left \{ a_{n} \right \}ln\, \, IR$ has a convergent subsequence ln R.