Let $f_n$ be a sequence of increasing and Riemann integrable function (i.e $f_1(x)\le...\le f_n(x) \ \forall x \in [a,b]$. If $f_n\to f$ pointwise and $f$ is Riemann integrable, show that $\lim_{n\to\infty}\int_{a}^{b}f_n(x)dx=\int_{a}^{b}f(x)dx$. If someone could give a hint how to prove the statement, I would really appreciate it.
What I'm trying to show is that $|\overline{S}_{\sigma}(f_n)-\overline{S}_{\sigma}(f)|<\epsilon$ and $|\underline{S}_{\sigma}(f_n)-\underline{S}_{\sigma}(f)|<\epsilon$ ($\overline{S}$ and $\underline{S}$ denote the upper and lower Darboux sum's and $\sigma$ is a subdivision).
If I succeed to show that, clearly $\lim_{n\to\infty}\int_{a}^{b}f_n(x)dx=\int_{a}^{b}f(x)dx$. But, I don't really see how to apply the pointwise convergence correctly and the fact that $f_n$ is increasing... I notice as well that $\lim_{n\to \infty}f_n(x)=f(x)=\sup(f_1,f_2,f_3,...)$ because $f_n$ is increasing. Thank you in advance for help!