Using the nested set property to prove uniform convergence

Question

Let $f_n$ be a continuous function on a compact set $E$

for each n, let $f_n$ convergence pointwise to $F$ which is again a continuous function

Suppose that for any $p \epsilon E$ the sequence $f_n(p)$ is an increasing sequence of real numbers. Prove that $f_n$ in fact converges uniformly on $E$

Hint:

Consider the set $C_n= (all \ p \epsilon E \ with \ F(p)-f_n(p)≥ \epsilon)$

I'm also given as a hint to use the nested set property which states that

$$\bigcap_k^ \infty C_k \ne \emptyset$$

So I believe then as $n \to \infty$ $C_n$ is not empty. But what does this prove? Any help please.

Answer 1

To be more precise, the nested set property says that if $\{ C_\alpha \}$ are a family of compact sets, and every intersection of a finite subfamily of $\{ C_\alpha \}$ is non-empty, then the infinite intersection is non-empty. Try and use the converse of this statement. It might be easier for you to approach this problem by simplifying the situation to a monotonically decreasing sequence of continuous functions $f_1(p) \geq f_2(p) \geq \dots$ for each $p$, with $f_i(p) \to 0$ pointwise for each point $p$, and proving that the $f_i$ converge uniformly to zero. Then generalize to your situation.