When does pointwise convergence imply uniform convergence? On an exam question (Question 21H), it is claimed that if $K$ is compact and $f_n : K \to \mathbb{R}$ are continuous functions increasing pointwise to a continuous function $f : K \to \mathbb{R}$, then $f_n$ converges to $f$ uniformly. I have tried proving this claim for the better part of an hour but I keep coming short. I suspect a hypothesis on equicontinuity has been omitted — partly because the first half of the question is about the Arzelà–Ascoli theorem — but I don't have access to the errata for the exam so I can't be sure.

Here is my attempted proof: let $g_n = f - f_n$, so that $(g_n)$ is a sequence of continuous functions decreasing pointwise to $0$. Clearly, $0 \le \cdots \le \| g_n \| \le \| g_{n-1} \| \le \cdots \le \| g_1 \|$, so we must have $\| g_n \| \longrightarrow L$ for some constant $L$. $K$ is compact, so for each $g_n$, there is an $x_n \in K$ such that $g_n(x_n) = \| g_n \|$, and there is a convergent subsequence with $x_{n_k} \longrightarrow x$ for some $x \in K$. By hypothesis, $g_n(x) \longrightarrow 0$, and by construction $g_{n_k} (x_{n_k}) \longrightarrow L$. I'd like to conclude that $L = 0$, but to do this I would need to know that the two sequences have the same limit. This is true if, say, $\{ g_n \}$ is an equicontinuous family, but this isn't one of the hypotheses, so I'm stuck.
The result is true as stated and is called Dini's Theorem.  A proof can be found in Chapter III, Section 1.4 of these notes or indeed in this wikipedia article.  (My notes are for a sophomore-junior level undergraduate class, so it is stated in the case in which $K$ is a closed interval.  But it is clear that the argument works for any compact topological space.)

(For some reason this is one of those named theorems that tends to be assigned as an exercise or come up on exams...)
It would appear that there is a slight error in your argument – there exist compact topological spaces that are not sequentially compact, such as $[0,1]^{[0,1]}$ (to become more familiar with such pathological spaces, I recommend the book Counterexamples in Topology). Here is a corrected (and much shortened) argument:

Let $g_n=f-f_n$, and note that $\max g_n$ is decreasing, and converges to $0$ iff $g_n\to0$ uniformly. If $\max g_n\ge M&gt;0$ for all $n$, then $\{g_n\ge M\}$ is a decreasing family of closed nonempty subsets of $K$, so by the finite intersection property, $\bigcap_n\{g_n\ge M\}$ is nonempty, which contradicts pointwise convergence.