Suppose that {fn} is a sequence of continuous functions on the interval [a, b] and that f is a continuous function on [a, b] such that {fn} converges uniformly to f on (a, b). Prove that {fn} converges uniformly to f on [a, b].
In other words, how can I demonstrate that {$f_n$} is also uniformly convergent at it's endpoints.
I know this Theorem: Suppose $f(x) = lim_{n→∞} f_n(x)$ for all $x ∈ [a,b]$, that each function $f_n$ is continuous on $[a,b]$, and that the convergence is uniform. Then $\int_a^b f(x)dx = lim_{n→∞} \int_a^b f_n(x)dx$.
But this theorem refers to a closed domain. Additionally, I know that $\int_a^a f(x)dx = 0$. I'm unsure if I'm on the right track.