Does pointwise convergence of continuous functions on a compact set to a continuous limit imply uniform convergence on that set?
I think yes.
Suppose that is a metric space, let K be a compact subset of M, and define such that there exists a continuous function with pointwise.
By definitions, such that
Now, what I want to show is that such that , so an epsilon that doesn't depend on x.
But I find it difficult to work through this, perhaps my understanding of uniform convergence is still poor. Am I on the right track thou?
Okay, here is my formal proof.
Claim: This sequence of functions converges pointwise to
For and , we would have .
Given , pick
For , we would have I'm stuck here, I know that this would converge to 0, but how would I prove that? What N I should pick to ensure this distance is less than epsilon?
Claim: This sequence of functions do not converge uniformly to 0.
Pick , then , whenever and for each index n, we will have But doesn't this still converges to 0? Did I pick the wrong x?
As I said, it's a tricky question.