Uniform convergence and pointwise convergence

Abstract

The aim of this material is to introduce the student to two notions of convergence for sequences of real-valued functions. The notion of pointwise convergence is relatively straightforward, but the notion of uniform convergence is more subtle. Uniform convergence is explained in terms of closed function balls and the new notion of sets absorbing sequences.

The differences between the two types of convergence are illustrated with several examples. Some standard facts are also discussed: a uniform limit of continuous functions must be continuous; a uniform limit of bounded functions must be bounded; a uniform limit of unbounded functions must be unbounded.

Target audience: Most of this material should be accessible to anyone who understands what a real-valued function is, and understands the notion of convergence of a sequence of real numbers. This should include most mathematics undergraduates by the end of their first year. An understanding of continuity and of boundedness for real-valued functions defined on various types of domain would help the student to understand the latter part of the material.