I'm studying my way through Apostol for a final. I have questions on Ex 10.38 through 10.42, but if I can get 10.38 I should be good for the rest.

$f\in L^2(I)$. The $L^2$-norm, $\|f\|$, is defined by the formula, $\|f\|=(\int\_I|f|^2)^{1/2}$.

10.38 If $\lim\_{x\to\infty}\|f\_n-f\|=0$, prove that $\lim\_{x\to\infty}\|f\_n\|=\|f\|$.

I was starting by applying the definition of norm to the integral, but I didn't know how to proceed from there. Are there some operations I can perform on the norm before putting the integral in there to simplify? If not, how do I manipulate this equation?

$\lim\_{x\to\infty}(\int\_I|f\_n-f|^2)^{1/2}=0$