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

...