This is my proof
Rewriting the proof (it's not very good)
Proof
Since $f$ is continuous, we have $\forall \epsilon > 0$, $\exists r > 0$ s.t.
$|f(x) - f(y)| < \epsilon$ whenever $\Vert x - y\Vert < r$
We are also told that $f(x) < C$ for some $C \in \mathbb{R}$
So if I go with the suggest that to choose $\epsilon = C - f(x)$, then I get
$|f(x) - f(y)| < C - f(x) \implies f(x) - C < f(x) - f(y) < C - f(x) \implies -C < -f(y) < C - 2f(x) \implies C - 2f(x) < f(y) < C$
Problem I have here is that I didn't use $r$ at all. I know I am doing something wrong, could someone point it out to me?