Pointwise convergence to uniform convergence

Does pointwise convergence of continuous functions on a compact set to a continuous limit imply uniform convergence on that set?

I think yes.

Proof.

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?

Thanks!