(a) Prove that $f_n(x) → f(x)$ uniformly on $E$ as $n → ∞.$ Let $E ⊂ R$ be a compact (i.e., closed bounded) set of real numbers. Suppose $\{f_n\}$ is a sequence of real-valued continuous functions which converges pointwise on $E$ to a function $f$ that is also continuous on $E.$ Suppose further that the sequence $\{f_n\}$ is monotonic: $f_{n+1}(x) ≤ f_n(x)$ for all $x ∈ E$ and all $n = 1,2,....$
(a) Prove that $f_n(x) → f(x)$ uniformly on $E$ as $n → ∞.$
(b) Show by example that the hypothesis of compactness is essential.

We have that each of the $f_n$ and the $f$ is uniformly continuous, which I thought may be useful, but am not exactly sure how.
Example that compactness is essential:

Take $f_n(x)=\chi_{[0,n]}$.

To do the original proof, have a look at Dini's theorem. (My proof would be exactly same as given here- so I didn't bother writing it out.)
Example that compactness is essential:

Take $f_n(x)=\chi_{[0,n]}$.

To do the original proof, have a look at Dini's theorem. (My proof would be exactly same as given here- so I didn't bother writing it out.)