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: 
Use these two facts:

* Every sequence has a monotone subsequence.
* Every bounded monotone sequence converges.
**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.