Let $f_{n}$ ($n=1,2,3,...$) and $f$ be functions defined on a set $E$.
The theorem states
Suppose $$\lim_{n\rightarrow \infty} f_{n}(x)=f(x) \quad (x \in E).$$ Put $$M_{n}=\sup_{x \in E} \mid f_{n}(x)-f(x) \mid.$$ Then $f_{n} \rightarrow f$ uniformly on $E$ if and only if $M_{n} \rightarrow 0$ as $n \rightarrow \infty$.
I don't understand why we need to assume pointwise convergence. Isn't convergence of the supremum simply equivalent to uniform convergence?
My take is this is a reflection of how you actually prove a sequence converges uniformly to some function. You of course need to find the pointwise limit first since that is the only candidate.