Proving continuous image of compact sets are compact How to prove: Continuous function maps compact set to compact set using real analysis?
i.e. if $f: [a,b] \rightarrow \mathbb{R}$ is continuous, then $f([a,b])$ is closed and bounded. 
I have proved the bounded part. So now I need some insight on how to prove $f([a,b])$ is closed, i.e. $f([a,b])=[c,d]$. From Extreme Value Theorem, we know that $c$ and $d$ can be achieved, but how to prove that if $c < x < d$, then $x \in f([a,b])$ ?
Thanks!

Lindsay, what you need is the intermediate value theorem, its proof is given in wikipedia.
