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.
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.