Let $\{f_n\}$ be a sequence of continuous functions on on $[0,1]$. Let $f_n \to f$ pointwise. If $f$ is continuous on $[0,1]$, is it true that $$\int_0^1 f_n(x) dx \to \int_0^1 f(x)dx?$$
I couldn't think of a counter-example, so my inclination is that it is true. If I can show that $f_n \to f$ uniformly, then I would be done, since I can choose an $N$ such that for all $n > N$ and $x\in [0,1]$ we have $|f_n(x) - f(x)| < \varepsilon$, which gives $$ \left| \int_0^1 f_n(x) - f(x) dx \right| \leq \int_0^1 |f_n(x) - f(x)|dx < \varepsilon$$
Can it be shown that $f_n \to f$ uniformly since we're working on a compact set and $f$ is continuous? Or can a counter-example be constructed from here?