>>1
If a sequence is cauchy in the sup norm that implies the sequence of functions converges uniformly.
Then use the definition of uniform convergence to proof that a sequence of bounded functions cannot converge to an unbounded one uniformly, which is pretty obvious.
I'll sketch a proof.
Ok, assume there exists a sequence -> f such that f is not bounded.
Now for any fn in the sequence there is a max |fn| , call this M.
However as f is unbounded there exists a point x s.t f(x) > M + 1
Therefore for all n sup|fn - f| > 1 at least. but then fn is not a cauchy sequence, contradiction.
Hmm, didn't use uniform convergence, oh well.