Show that if $f$ is continuous on $[0, \infty)$ and uniformly continuous $[a, \infty)$ for some positive constant $a$, then $f$ is uniformly continuous on $[0, \infty)$.
Here is my attempt at the proof. A classmate is concerned my proof leaves a gap around $a$ that is not rigorously defined.
Let $f: [0, \infty) \rightarrow \Bbb R$ continuous and let $f: [a, \infty) \rightarrow \Bbb R$ uniformly continuous for $a>0$
Consider the interval $I:= [0,a]$ and $[0,a] \subseteq \Bbb R$
Note the interval $[0,a]$ is closed and bounded. By the Heine-Borel Theorem, since $I$ is closed and bounded, $I$ is compact.
If a function $f$ is continuous on a compact space then it is uniformly continuous thus $f$ is uniformly continuous on $[0, \infty)$
This is a proof for my real analysis class and I've used a proof that I learned last semester in topology. So it may be the case that this technique is not allowable. Can anyone advise?
Is this proof sufficient to confirm that the points around $[0,a)$ are continuous?