I am trying to prove:
$f$ is a real, uniformly continuous function on the bounded subset $E$ in $\mathbb{R}^1 \implies f$ is bounded on $E$.
My attempt:
When I wrote this proof, I thought that the proof was correct. However, a friend of mine pointed out that the last paragraph of the proof is not correct (didn't exactly specify what's incorrect, just that something is). Can someone please point out what's incorrect about this proof, if at all?
That proof is wrong because the domain of $f$ is $E$. You can restrict $f$ to a subset of $E$, but, since $\overline E$ can be strictly bigger than $E$, it makes no sense to talk about $f\left(\overline E\right)$.
It actually follows from the fact that $f$ is uniformly continuous that you can extend $f$ to a continuous function whose domain is $\overline E$, but that is not obvious at all.