Math Help - convergence in L^2

1. convergence in L^2

Let $f_n \rightarrow f$ in $L^2(X,F,m)$.
If $m(X)<\infty$, then $\int_X f_n \rightarrow \int_X f$.
any hint would be appreciated.

2. Originally Posted by Archi
Let $f_n \rightarrow f$ in $L^2(X,F,m)$.
If $m(X)<\infty$, then $\int_X f_n \rightarrow \int_X f$.
any hint would be appreciated.
Cauchy–Schwarz (or Hölder): $\Bigl|\int_X f_n - \int_X f\Bigr| \leqslant \int_X|( f_n-f)*1| \leqslant \Bigl(\int_X |f_n-f|^2\Bigr)^{1/2} \Bigl(\int_X 1^2\Bigr)^{1/2}$.