Verify a Proof about L2 Norm

I'm working through some suggested problems and one asks to show that if uniformly on a compact interval , and each continuous on , then , where ||...|| is the norm. I first want to verify that the following is sound: If then . My main worry is moving the limit outside of the square root in this argument.

Anyway, given that this is sufficient, then we know that for sufficiently large , for all , and since is compact it's therefore bounded, let's say the length of the interval is bounded by . So . But at no point did I seem to use continuity of ...