If $f_n \to 0$ in $L^2$ on $[0,1]$, show that $f_n\to 0$ in $L^1$ on $[0,1]$, providing that $f_n$ is continuous for each $n$.
I'm feeling really stupid right now, I can't seem to figure this out. So far, I've written:
$$\int_0^1 |f| = \int_0^1(|f|^{1/2})^2 \le ||f^{1/2}||_2 ||f^{1/2}||_2,$$ but that just gets me right back to where I started. What am I missing?