Show that if $f_n$ converges to f in uniform metric then $f_n$ converges to f in $L^2$ metric.
Attempt at solution:
If $f_n$ converges to $f$ in uniform metric then $\sup_{x \in I}|f_n(x)-f(x)| \to 0$), then $$\|f_n-f\|_2=\int_a^b |f_n(x)-f(x)|^2\,dx \leq \int_a^b sup (|f_n(x)-f(x)|)^2\,dx\ \leq \ (b-a) sup (|f_n(x)-f(x)|)^2\,dx\ $$ and we know this last equality tends to 0, so we have $L^2$ convergence. I'm not sure if my proof is entirely correct however. Any help would be much appreciated.