"Let be functions defined on converging uniformly to a function f. IF each is bounded, say by , THEN f is bounded."

converges to f uniformly =>|| ||∞ ->0 as k->∞

Also, we know| |≤ for all k, for all x.

But why does this imply that f is bounded? I don't see how to prove it.

Also, why do we need uniform convergence? (why is pointwise convergence not enough?)

Any help is appreciated!

[also under discussion in math links forum]