# Thread: Uniform Convergence & Boundedness

1. ## 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]

2. Originally Posted by kingwinner
"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]
Hint:
Spoiler:
$\displaystyle \|f\|_{\infty}-\|f_n\|_{\infty}\leqslant \|f-f_n\|_{\infty}$

3. Originally Posted by Drexel28
Hint:
Spoiler:
$\displaystyle \|f\|_{\infty}-\|f_n\|_{\infty}\leqslant \|f-f_n\|_{\infty}$
OK, so if we choose n large enough, we will have ||f||∞ ≤ ε + $\displaystyle A_k$. To show f is bounded, I think we can just fix some ε.

Why is pointwise convergence not enough?

4. Originally Posted by kingwinner
OK, so if we choose n large enough, we will have ||f||∞ ≤ ε + $\displaystyle A_k$. To show f is bounded, I think we can just fix some ε.

Why is pointwise convergence not enough?
Well, you need to be careful. $\displaystyle A_k$ depends on $\displaystyle f_k$, right?

5. Originally Posted by Drexel28
Well, you need to be careful. $\displaystyle A_k$ depends on $\displaystyle f_k$, right?
Why does it matter? What should we do instead? Can we take the maximum over all the A_k and use that as a bound?