First off, I apologize for the bad wording; I am new to real analysis, and I have a few questions regarding uniform convergence.
(1)
We know that if $f:E\rightarrow \mathbb{R}$ and $f_n\rightarrow f$ uniformly $\forall x \in E$,
$\lim_{x \to p} \lim_{n \to \infty} f_n(x) = \lim_{n \to \infty} \lim_{x \to p} f_n(x)$.
Is there a particular name for this theorem? $ $
(2)
If $\lim_{x \to p} \lim_{n \to \infty} f_n(x) = \lim_{n \to \infty} \lim_{x \to p} f_n(x)$, does this imply uniform convergence; If we can 'switch around' limits, does this imply that $f_n$ is uniformly convergent?
If such a similar question/post exists, please feel free to redirect/criticize me. Thank you a lot for all the help.