I have the following question: Let $(f_n)_n$ be a sequence of continuous functions on $[a,b]$, such that $f_n$ converges pointwise to $f$, where $f$ itself is continuous, does then $$\int_a^b f(x)dx= \lim_{n \rightarrow \infty} \int_a^bf_n (x) dx$$ hold?
I have been trying to prove this, but this didn't lead to anywhere. When I tried to come up with counterexamples, all examples did satisfy $$\int_a^b f(x)dx= \lim_{n \rightarrow \infty} \int_a^bf_n (x) dx.$$
Does anyone know a proof/counterexample?