15 - Bolzano-Weierstrass Theorem

Bolzano-Weierstass Theorem

Let $(a_n)$ be a bounded real sequence then

$$(a_n)\text{ has a convergent subsequence}$$

Proof

By the Scenic Viewpoints Theorem, $(a_n)$ has a monotone subsequence
As the monotone subsequence is bounded (by assumption of whole)
Then by Monotone Subsequences Theorem, the subsequence converges