Solution
= Solution
Fix $x$ and $ε>0$. Since $\mathcal K$ is compact in the <uniform norm>, choose a finite $ε/3$-net $f_1,\ldots,f_r$. By <continuous function>[continuity], there is a neighbourhood $U$ of $x$ on which $|f_i(y)-f_i(x)|<ε/3$ for every $i$. For $f$ in $\mathcal K$, choose $f_i$ with $\|f-f_i\|_∞<ε/3$. The <triangle inequality> gives $|f(y)-f(x)|<ε$ on $U$, uniformly in $f$. Thus $\mathcal K$ is <equicontinuous family of functions>[equicontinuous].