A sequence of functions converges uniformly to a function
on a set
if for each
there is an integer
such that for
for all
and
We say thatis uniformly convergent onwith limit function
It is important to note thatmay depend on
Uniform convergence is illustrated below.
Example: Prove that the complex function
converges uniformly on
to the function
For each
so
as
Moreover
Chooseso that
for n>N and clearly
Hence
converges uniformly to f on E.
This example illustrates a strategy for proving uniform convergence.
-
Determine the limit function f by evaluating
for
-
Find a null sequence
of positive terms such that
for
and all