Exercise 7.2

Let $𝜀>0$. By Theorem 7.8 there is an integer $N$ such that for all $n,m\ge N$ and $x\in E$,

Hence for all $n,m\ge N$ and $x\in E$,

Hence, also by Theorem 7.8, $\{f_n+g_n\}$ converges uniformly on $E$.

If $\{f_n\}$ and $\{g_n\}$ are uniformly convergent sequences of bounded functions, then by Exercise 1 they are uniformly bounded. That is, there is a number $M$ such that $|f_n(x)|<M$ and $|g_n(x)|<M$ for all $n$ and for all $x\in E$. By Theorem 7.8 there is an integer $N$ such that for all $n\ge N$ and for all $x\in E$ we have

Hence for all $n,m\ge N$ and $x\in E$,

Hence, also by Theorem 7.8, $\{f_n{g}_n\}$ converges uniformly on $E$.