A sequence of functions converges uniformly to a functionon a setif for eachthere is an integersuch that forfor alland
We say thatis uniformly convergent onwith limit functionIt is important to note thatmay depend onUniform convergence is illustrated below.
Example: Prove that the complex functionconverges uniformly onto the function
Chooseso thatfor n>N and clearly
Henceconverges uniformly to f on E.
This example illustrates a strategy for proving uniform convergence.
Determine the limit function f by evaluatingfor
Find a null sequenceof positive terms such thatfor and all