What does it mean when someone asks you to show that a series of functions converges uniformly without specifying to what function? In the past when I've delt with uniform convergence, I've always been asked to show that a sequence converges uniformly to a given function. But what does the question mean when no limit-function is specified?
For example: "Show that $\sum_{n=0}^\infty a_n f_n(x)$ converges uniformly". What does this mean? What am I asked to do in such cases?
If a sequence of functions converges uniformly to a function, then it also converges pointwise to that function. So you can find the pointwise limit, then check that it converges uniformly to that limit.