1. ## analysis

$\mbox {show that if a sequence has a bounded subsequence , it has a convergent subsequence$

2. Originally Posted by flower3
$\mbox {show that if a sequence has a bounded subsequence , it has a convergent subsequence$
Here is an outline of a proof.
We may as well prove this statement: Every bounded sequence contains a convergent subsequence.

Lemma 1, Every sequence, $\left(x_n\right)$, contains a monotone subsequence.
Easy proof: Let $S=\{n:~\forall j>n,~x_j>x_n\}$. Two cases: $S$ is infinite or finite.
In the first case the subsequence is increasing.
In the second case the subsequence is non-increasing.

Lemma 2, Every bounded monotone sequence converges.

3. I assume that you mean a sequence in $\mathbb{R}$, since this need not be true in other metric spaces.