Problem:
Let $f: \mathbb{R}^n \to \mathbb{R}$ a continuous function, so that for every $\epsilon >0$, there exits a compact set $K \subset \mathbb{R}^n$ with $|f(x)|<\epsilon$ for all $x \in \mathbb{R}^n-K$.
Show that:
(i) $f$ takes on $\mathbb{R}^n$ a global maximum and minimum.
(ii) $f$ is uniformly continuous.
Questions/Issues:
Currently struggling to figure out the beginning of the proof. For what I know, there are compact sets that restricts $f$ to a $\epsilon$. So from what I know intuitively, that there is at least a point, at which $f$ is bounded to a compact set.
For the uniform continuity, I currently have no idea, where the beginning is to solve that part of the problem.
Thanks for the replies in advance.