Here is rudin's statement
Theorem 7.9 suppose $$\lim_{n\to\infty}f_n(x)=f(x)(x\in E).$$ Put $$M_n=\sup_{x\in E}|f_n(x)-f(x)|.$$ Then $f_n \to f$ uniformly on $E$ if and only if $M_n \to 0$ as $n \to \infty$.
Here is my question
Is the first condision unnecessary? Can I just point out ${f_n}$, $f$ and $x \in E$ instead?