Exercise 7.16

Let $𝜀>0$ and let $\delta >0$ such that if $d(x,y)<\delta$ then $|f_n(x)-f_n(y)|<𝜀∕3$ for all $n$. Let $x\in K$ and let $N_x$ be an integer large enough so that if $m>N_x$ and $n>N_x$, then $|f_m(x)-f_n(x)|<𝜀$. Then for all $m>N_x$ and $n>N_x$ and all $y\in K$ such that $d(x,y)<\delta$, we have

Since $K$ is compact, there are a finite number of points $x_1,\dots ,x_M$ in $K$ such that the neighborhoods of radius $\delta$ centered at the $x_i$ cover $K$. So if we let $N=\max <!--\; FUNCTION\; APPLICATION\; -->(N_{x_1},\dots <!--\; FUNCTION\; APPLICATION\; -->,N_{x_M})$, then for $m>N$ and $n>N$ and for all $y\in K$, we have $|f_m(y)-f_n(y)|<𝜀$. Hence by Theorem 7.8, $\{f_n\}$ converges uniformly on $K$.