Let $\forall n \in \mathbb{N}$ there is continuous function $f_n:[a, b]\rightarrow \mathbb{R}$ and let's suppose that sequence $\sum_{n\ge 1}\max_{x\in[a,b]}|f_n(x)|$ converges.
Show that the function $f(x) := \sum_{n\ge1}f_n(x)$ is correctly stated and that it is continuous on $[a, b]$
Please, can you give me a hint how to start the solution and I don't quite understand what means "to prove that function is correctly stated"
Edit: Ok, I can prove that it is correctly stated (the proof of it is exactly like @avs's answer. What about continuity? I think that I should prove that $\int_a^bf(x)$ is continuous
For each $x \in [a, b]$, the series $\sum_{n \geq 1} |f_{n}(x)|$ is majorized by the convergent series $\sum_{n \geq 1} \max_{[a, b]} |f_{n}|$, hence converges.
Therefore, the series $\sum_{n \geq 1} f_{n}(x)$ converges absolutely. Therefore, by the completeness of ${\mathbb R}$ in the metric induced by $| \cdot |$, the series $\sum_{n \geq 1} f_{n}(x)$ converges.