Exercise 7.1

Let $\{f_n\}$ be a uniformly convergent sequence of bounded functions on a set $E$. For each $n$, there is a number $M_n$ such that $|f_n(x)|<M_n$ for all $x\in E$. By Theorem 7.8, there is an integer $N$ such that $|f_n(x)-f_N(x)|<1$ if $n\ge N$ for all $x\in E$. Let $M=\max <!--\; FUNCTION\; APPLICATION\; -->\{M_1,\dots <!--\; FUNCTION\; APPLICATION\; -->,M_N\}$. Then for $n\le N$ and $x\in E$ we have $|f_n(x)|<M+1$, and for $n\ge N$ and $x\in E$ we have

That is, the $f_n$ are uniformly bounded by $M+1$ in E.