(a) For each $x$ let $n_{x}$ be such that $f(x)-f_{n_{x}}(x)<\epsilon$ . Now, since $f(x)-f_{n_{x}}(x)$ is continuous in $x$ for the fixed $n_{x}$ , there is an open neighborhood $U_{x}$ of $x$ such that for each $x'\in U_{x}$ : $f(x')-f_{n_{x}}(x')<\epsilon$ . Since $f_{n}$ is monotone increasing, for all greater $n$ this still holds. Moreover, all $U_{x}$ ’s cover $X$ and, since $X$ is compact, there is a finite subcover $U_{x_{i}}$ . Let $N$ be the maximum of $n_{x_{i}}$ . Then for each $x\in X$ there is some $x_{i}$ such that $x\in U_{x_{i}}$ , and for all $n\ge N\ge n_{x_{i}}$ : $f(x)-f_{n}(x)<\epsilon$ .(b) The example of Exercise 9 of §21 can be restricted to the compact domain $[0,1]$ . $\arctan(x+n)$ is a monotone increasing sequence converging to a constant function but not uniformly (for any $n$ : $f_{n}(-n)=0$ ).