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 $\displaystyle (M,d)$ is a metric space, let K be a compact subset of M, and define $\displaystyle f_n:K \rightarrow \mathbb {R} \ \ \ \forall n \in N$ such that there exists a continuous function $\displaystyle f:K \rightarrow \mathbb {R} $ with $\displaystyle f_n \rightarrow f $ pointwise.

By definitions, $\displaystyle \forall \epsilon > 0, \exists M_{( \epsilon ,x)} $ such that $\displaystyle d(f_n,f)< \epsilon \ \ \ n \geq M $

Now, what I want to show is that $\displaystyle \exists M_{ \epsilon } $ such that $\displaystyle d(f_n,f) < \epsilon \ \ \ n \geq M$, 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!