Uniform Convergence & Boundedness

"**Let $\displaystyle f_k$ be functions defined on $\displaystyle R^n$ converging uniformly to a function f. IF each $\displaystyle f_k$ is bounded, say by $\displaystyle A_k$, THEN f is bounded.**"

$\displaystyle f_k$ converges to f uniformly =>||$\displaystyle f_k - f$||∞ ->0 as k->∞

Also, we know|$\displaystyle f_k(x)$|≤ $\displaystyle A_k$ 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]