Can someone point me to a detailed proof of this, or give it?

The stuff I can find online alludes to the triangle inequality, but I can't seem to join the dots...

if is a Cauchy sequence of continuous functions I need to show that they converge (uniformly) to some function g (so that g is continuous), right? So the triangle inequality gives

the first term on the right is arbitrarily small for n & m big enough... but what about the second one? Or am I going about this all wrong?