Theorem 7.11: Suppose $f_n \to f$ uniformly on a set $E$ in a metric space. Let $x$ be a limit point of $E$, and suppose that $$\lim_{t\to x} f_n(t) = A_n \qquad (n \in N).$$ Then $\{A_n\}$ converges, and $$\lim_{t\to x} f(t) = \lim_{n\to \infty} A_n.$$
In other words, the conclusion is that
$$\lim_{n\to\infty}\lim_{t\to x}f_n(t)=\lim_{t\to x}\lim_{n\to \infty}f_n(t)$$
Theorem 7.12: If $(f_n)$ is a sequence of continuous functions on $E$ and if $f_n \to f$ uniformly, then $f$ is continuous on $E$.
I guess I'm having an issue understanding what theorem 7.11 is actually saying in English? Does the conclusion mean that the limit function of $(f_n)$ is continuous at any limit point of $E$? Why is it that continuity for a limit function looks like
$$\lim_{n\to\infty}\lim_{t\to x}f_n(t)=\lim_{t\to x}\lim_{n\to \infty}f_n(t)$$
and not something more like
$$
\lim_{t \rightarrow x} f(t) = f(x).
$$
Why does it follow naturally from 7.11 that 7.12 holds? I feel like I'm missing a simple understanding of what theorem 7.11 is actually saying, although I feel like I understand the proof (which is probably wrong to say because I don't understand the conclusion). Any insights are greatly appreciated.
Best Answer
Since you requested an explanation in plain English, the first theorem says that when convergence is uniform, the order in which you take the limits of the function $f_n$ does not matter. The ability to interchange limit operations in this way is one of the reasons why uniform as opposed to mere pointwise convergence of a sequence of functions is viewed as advantageous. The second theorem is just a special case of the first insofar as 7.11 says nothing about the functions being continuous. The most important piece of information that is being communicated in what you have written is that "the uniform limit of a sequence of continuous functions is continuous." This is a fundamental result that is routinely invoked in proofs in analysis, and so it is a good idea to not only memorize it, but to also develop a clear understanding of the intuition that underlies it.