Suppose {$f_n$}, {$g_n$} defined on $E$ and,
(a) $\Sigma f_n$ has uniformly bounded partial sums; (b) $g_n \to 0$ uniformly on $E$ (c) $g_1(x)\geq g_2(x)\geq g_3(x)\geq ...$ for every $x \in E$.
Prove that $\Sigma f_n g_n$ converges uniformly on $E$.
My solution:
By uniform convergence of $g, \exists N\in \Bbb N$ s.t.$|g_n|\leq \epsilon/q$ for $n\geq N$ and $q$ being the upper bound on $\Sigma f_n$.
By Cauchy Criterion for convergence and choosing $M,N \geq N$, we have
$|\Sigma_{M+1}^{N} f_k g_k|\leq \Sigma_{M+1}^{N} |f_k g_k|\leq \Sigma_{M+1}^{N} |f_k| |g_k|\leq |g_k|\Sigma_{M+1}^{N} |f_k| \leq \epsilon q/q= \epsilon$,
Is the solution correct? Please point out the flaws. Thanks