If $\sum|g_n|$ converges, then $\sum |f_ng_n|$ converges

Question

I'm trying to do this question:

If $\sum|g_n|$ converges uniform in X and there is $k\gt 0$ such that $|f_n(x)|\leq k$ for all $n\in \mathbb N$ and all $x\in X$, then $\sum |f_ng_n|$ converge uniform in X.

I can't solve this question, I don't know why because this one seems very simple, I need a hint or something to begin to solve this question

Thanks a lot

Answer 1

Hint: for a fixed integer $N$, what is a good upper bound of $$\sup_{x\in X}\sum_{n\geqslant N}|g_n(x)f_n(x)| \, ?$$

Answer 2

$$\left|\sum_{i=n}^mf_i(x)g_i(x)\right|\le\sum_{i=n}^m|f_i(x)||g_i(x)|\stackrel{\forall\,x\in X}\le k\sum_{i=n}^n|g_i(x)|$$

Now, for any $\,\epsilon > 0\,$ there exists $\,N_\epsilon\in\Bbb N\,$ .s.t

$$n,m>N_\epsilon\Longrightarrow \sum_{i=n}^m |g_i(x)|<\frac{\epsilon}{k}\,\,,\,\forall\,x\in X\;\;,\;\;\text{so }\ldots$$