I would appreciate if you gave my proof a look over, i'm getting a bit stuck towards the end. thank you very much. first time writing in latex by the way, so apologies beforehand.
claim: let $f: M \to N$ be a uniformly continuous function. Show that $f$ maps bounded sets to bounded sets.
proof: If $ m \subset M$ is bounded, then $ d(p,q) < B $ for fixed p and all q $ \in $ m.
let $F (m) $ be the image of m.
by definition of uniform continuity,
For all $ \epsilon > 0$ there exists $\delta > 0$ such that for all p,q such that $d(p,q) < \delta $ $\implies $ $d(f(p),f(q)) < \epsilon$
edit: cannot let $\delta = m$
so now i'm thinking that if we hold p and vary q, we will cover M with a set of open balls. Because M is bounded, that means this set of open balls is finite. and we can use that to similarly bound $f(m)$ with a finite set of open balls by the continuity definition.
*** this strategy would be assuming the set were "totally bounded", meaning it can be covered by a finite number of open balls of any radius. thanks for pointing out the term, user251257
edit: it's two general metric spaces, we cannot assume it's R.
Best Answer
In general this is not true for arbitrary metric spaces but if $M=N=\mathbb R$ with the standard metric, you can argue as follows:
$f$ is uniformly continuous on $m\subseteq M$, so it extends to a (uniformly) continuous function on $\overline m$, which is also bounded, hence compact. Of course, $f(m)\subseteq f(\overline m)$, and the latter is compact hence bounded,so that $f(m)$ is bounded as well.