I have that $f(x) : \mathbb{R}_+ \rightarrow \mathbb{R}$ is a uniformly continuous function s.t. $f(0) = \alpha \in \mathbb{R}_+$ and $lim_{x \rightarrow \infty} f(x)=0.$
I believe that if I can show that $\mathbb{R}_n$ is bounded, then I can use the extreme value theorem to show that the maximum must exist and be in $\mathbb{R}_+$. My idea is that I can define my distance function $d(x,y) = |x-y|$ and create a complete metric space. Thus my uniformly continuous function maps from a complete metric space to $\mathbb{R}$, making my function bounded. Since my function is uniformly continuous and bounded on a closed set $\mathbb{R}_+$, there exists $c$ and $d$ in $[0, \infty)$ such that:
$f(c) \leq f(x) \leq f(d) \forall x \in [0, \infty)$ with $d$ being my maximum.