For the past two weeks, I've tried to prove two different results that hold the same structure:
Suppose a property that holds for a dense subset of a metric space. Prove that it holds for the entire metric space.
In this case, both questions were related to uniform convergence:
i) Show that if $(f_{n})$ is a sequence of continuous functions on M, and if $\sum_{n=1}^{\infty}f_{n}$ is converges uniformly on a dense subset A of M, then the series converges uniformly on all of M.
ii) Let A be a dense subset of M. If $(f_{n})$ is a sequence of continuous functions on M, and if the sequence converges uniformly on A, prove that $(f_{n})$ converges uniformly on M.
My questions are the following:
What is the framework I should have in my mind when trying to prove these kinds of properties? What is the basic frame of mind you have when proving such results?
Because I feel that I'm missing something, that is, my intuition about how a dense subset works is not strong enough.
By the way, I'm sorry for the wall of text :)
Thanks in advance!