Proving if L2 converges, then L1 as well.

Question

I would like to know how to prove that since having $$ \left(\int_{a}^{b}|f(x)_{n}-f(x)|^2 dx\right)^{\frac{1}{2}} \rightarrow 0 $$ when $n \rightarrow\infty $. Then we have that $$ \left(\int_{a}^{b}|f(x)_{n}-f(x)| dx\right) \rightarrow 0$$ also when $n \rightarrow\infty $. How may I solve this just by bounding the $L1$ distance. Thanks in advance.

Answer 1

\begin{align*} \int_{a}^{b}|f_{n}(x)-f(x)|dx\leq\left(\int_{a}^{b}|f_{n}(x)-f(x)|^{2}dx\right)^{1/2}\left(\int_{a}^{b}1^{2}dx\right)^{1/2}. \end{align*}