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

ANSWER :

Proof :  

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

Let be a bounded sequences IR. Then for some To prove applying contradiction .

Assume contains no convergend subsequence , then such that  

Contains finitely many .

Otherwise if there excused an x such that and finitely many .

would have a convergent subsequence contaning.

Then,

Then by HBT a finite subcover such that ,

Therefore, is contained subcover.

There we conclude that

has finitely marry.

Here, has a convergent subsequence ln R.