Dini's Lemma

In our last project, we use Dini's lemma in one step of proof. This result, although should be part of undergraduate analysis, is really like magic. It says "the monotone convergence of continuous function $\{f_n\}_{n \in \mathbb{N}}$ to a continuous limit $f$ will give us locally uniform convergence. "

The monotone convergence is the key. For the case of decreasing sequence, we fix $N$, then for all $n > N$, then $f_n(y) - f(y) \leq f_N(y) - f(y)$. We then apply the classcial triangle inequality for a $\delta$-neighbor of $x$ for both $f$ and $f_N$ that 

$$f_n(y) - f(y) \leq f_N(y) - f(y) \leq \vert f_N(y) - f_N(x)\vert  + \vert f_N(x) - f(x)\vert + \vert f(x) - f(y)\vert.$$

Then only the information of $f_N$ and $f$ near $x$ can control all the convergence of sequence $n \geq N$.