Uniform Convergence & Boundedness

1. Apr 12, 2010

kingwinner

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

fk converges to f uniformly =>||fk - f|| ->0 as k->∞
Also, we know|fk(x)|≤ Ak for all k, for all x

But why does this imply that f is bounded? I don't see why it is necessarily true.

Any help is appreciated!

Last edited: Apr 12, 2010
2. Apr 12, 2010

mathman

|f|=|f-fk+fk| ≤ |f-fk| + |fk|≤ ε + Ak

You need uniform for ε to be usable for all x.

3. Apr 13, 2010

kingwinner

Why does it imply f is bounded? Is ε fixed here?

Also, why do we need uniform convergence? (why is pointwise convergence not enough...I still don't see why)

Can someone explain this, please? Thanks!

4. Apr 13, 2010

mathman

Since the norm is defined to be the sup over all values of x
We can write:
||f||=||f-fk+fk|| ≤ ||f-fk|| + ||fk||≤ ε + Ak
Therefore ||f||≤ ε + Ak, which is a bound. ε will depend on k, but not on x.

The uniform convergence means that we can use one ε to be a bound for ||f-fk||. If it was simply pointwise convergence, ε would be a function of x and might not be bounded.

Last edited: Apr 13, 2010
5. Apr 13, 2010

kingwinner

Sorry, I don't think it makes sense. ε is supposed to be given, always. And we're supposed to find N such that for k>N, ... will be < ε

Also, Ak depends on fk. What we want is a bound M which is a constant and does not depend on k, right?

Last edited: Apr 13, 2010
6. Apr 14, 2010

Landau

You need to show that f is bounded, i.e. find some positive real number M such that for all x we have |f(x)|<M. Here M cannot depend on x, but it may ver well depend on k (why not?).

mathman found such M in his first post.

7. Apr 14, 2010

evagelos

put ε =1 and k = N+1

8. Apr 14, 2010

mathman

(Uniform convergence) For every ε > 0 there exists an N so that for all k > N, ||f-fk|| < ε.
(bounded) ||fk|| < Ak. (Use any k > N)

Therefore ||f|| < ε + Ak. Thus, in plain English, f is bounded!! This proof does not determine the value of the minimum bound.