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.)