# Extending uniform convergence from open intervals to the entire (compact) interval

I'm a self-studier. This is the essence of a question from a homework set I saw.

Given that a sequence of real-valued functions f(n) converges uniformly on each open interval of a finite subcover of [a,b], show that it converges uniformly on [a,b].

My question what is the issue that has to be resolved to complete the proof.

(I saw a question that started in a similar vein, but didn't end up there.)

Thanks.

-

Hint: if something is true for all $n > N_i$ when $x \in U_i$, then it's true for $n > \max_i N_i$ when $x \in \bigcup_i U_i$.