"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]