Let $(X,d)$ be a metric space. Let $f_n : X \rightarrow \mathbb{R} $ be continuous for each $n \geq 1$. Assume that $|f_n(x)|\leq a_n$ and assume that the series $\sum_n a_n$ converges. Show that $F(x) = \sum_{n=1}^\infty f_n(x) $ defines a continuous function.
My attempt: Since $|f_n(x)|\leq a_n$ for all $n$ and since $\sum_n a_n$ converges, we know that $\sum f_n(x)$ converges.
What do I do from here?