Definition. The sequence $\{f_n(x)\}$, $n\in \mathbb{N}$ defined on an interval $I$ converges in mean-square to the function $f(x)$ on $I$ if $\lim_{n\to \infty} \int_I {|f_n(x)-f(x)|}^2\, dx =0$. We write $f_n(x)\to f(x)$ in mean-square on $I$ as $n\to \infty$.
Theorem.
(a) If $f_n(x)\to f(x)$ in $L^\infty$ on a finite interval $I$, then $f_n(x)→f(x)$ in $L^2$ on $I$.
(b) If $f_n(x)\to f(x)$ in $L^2$ on a finite interval $I$, then $f_n(x)\to f(x)$ in $L^1$ on $I$.
I need proof for theorem please.
For (a): Note that $$\int_I|f_n-f|^2\le \|f_n-f\|_\infty^2 \lambda(I),$$ where $\lambda(I)$ is the length of the interval.
For (b): Note that $$\int_I|f_n-f|=\int_I(|f_n-f|\cdot 1)\le \|f_n-f\|_2 \lambda(I)^{1/2}.$$