Question: How to prove that the sequence $\{a\_n\}$ has no convergent subsequence if and only if $|a\_n|$ approaches infinity?
The forward direction is obvious, how to prove the other direction?

Answer: 
First show that, if $|a\_n|$ doesn't approach $\infty$, then $a\_n$ has a bounded subsequence; then invoke the fact that a bounded sequence of real numbers has a convergent subsequence.
**Hint** If this result not true then [weierstrass sequence theorem](http://en.wikipedia.org/wiki/Bolzano%E2%80%93Weierstrass_theorem) is false. 

**Up date** Imitate this [proof](http://en.wikipedia.org/wiki/Bolzano%E2%80%93Weierstrass_theorem#Proof) of [weierstrass sequence theorem](http://en.wikipedia.org/wiki/Bolzano%E2%80%93Weierstrass_theorem) (with necessary adaptations of course) to get the proof of this fact. What I mean is that the same idea of this proof of weierstrass sequence theorem can be used to prove this result.