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Uniform Convergence of Function Sequences

By Michael Haag, Richard Baraniuk

Uniform Convergence of Function Sequences

For this discussion, we will only consider functions with g n g n where RR

Definition 1: Uniform Convergence
The sequence g n | n =1 n 1 g n converges uniformly to function gg if for every ε>0 ε 0 there is an integer NN such that nN n N implies
| g n tgt|ε g n t g t ε
1
for all tR t .
Obviously every uniformly convergent sequence is pointwise convergent. The difference between pointwise and uniform convergence is this: If g n g n converges pointwise to gg, then for every ε>0 ε 0 and for every tR t there is an integer NN depending on εε and tt such that Equation 1 holds if nN n N . If g n g n converges uniformly to gg, it is possible for each ε>0 ε 0 to find one integer NN that will do for all tR t .

Example 1

t ,tR: g n t=1n t t g n t 1 n Let ε>0 ε 0 be given. Then choose N=1ε N 1 ε . Obviously, n ,nN:| g n t0|ε n n N g n t 0 ε for all tt. Thus, g n t g n t converges uniformly to 00.

Example 2

t ,tR: g n t=tn t t g n t t n Obviously for any ε>0 ε 0 we cannot find a single function g n t g n t for which Equation 1 holds with gt=0 g t 0 for all tt. Thus g n g n is not uniformly convergent. However we do have: g n tgt pointwise g n t g t pointwise

Conclusion:
Uniform convergence always implies pointwise convergence, but pointwise convergence does not guarantee uniform convergence.