real analysis - Proving Uniform Continuity

I'd like to prove that If $f$ is continuous on $[a, \infty)$, and $\lim_{x \to \infty} f(x)< \infty$, then $f$ is uniformly continuous on $[a, \infty)$.

My book contains a lot of theorems that have to do with proving uniform continuity, but all of them require the set to be closed and bounded. Any help would be appreciated!