Uniform Convergence

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

For eachsoas


Chooseso thatfor n>N and clearly

Henceconverges uniformly to f on E.

This example illustrates a strategy for proving uniform convergence.

  1. Determine the limit function f by evaluatingfor

  2. Find a null sequenceof positive terms such thatfor and all