Proving that if $\sum_0^\infty f_n$ converges uniformly on $D$ to f and $f_n$ is bounded, then f is bounded.

Question

"Prove that if each $f_n$ is a bounded function and $\sum_0^\infty f_n$ converges uniformly on $D$ to $f$, then $f$ is a bounded function"

I don't know how to do this at all. Any help appreciated.

Answer 1

By Cauchy's Convergence Principle of uniformly convergence, $\forall\varepsilon>0\exists N>0$ s.t.$\forall n,m>N$,$|f_n-f_m|<\varepsilon,\forall x\in D$. In particular, $|f_n-f_{N+1}|<\varepsilon,\forall x\in D$. Thus $|f_n|<sup|f_{N+1}|+\varepsilon$ i.e. $f_n$ is uniformly bounded. Then you can use the definition of uniformly convergence on $f$.