Let $\sum|g_n|$ converge uniformly in $X$. Suppose there exists $K$ such that $|f_n(x)|\leq K$, for all $n\in\mathbb{N}$ and all $x\in X$. Prove that $\sum f_ng_n$ converges absolutely and uniformly in $X$.
How can I do this problem, it's from a book that I'm using of real analysis, can someone help me with the solution D:? It use a trick or something?