Metric Spaces

Abstract

This book has discussed at length how one writes proofs about the limits and continuity of functions whose domains and ranges are subsets of the real numbers, $\mathbb{R}$. Although the real numbers is a far simpler set to study than many other naturally arising sets in Analysis, the techniques learned while dealing with real-valued functions of a real variable can be applied almost exactly to prove similar theorems about functions defined on other domains with other types of ranges. It is instructive to take note of the properties of the real numbers that play important roles in these proofs. In particular, most of the proofs about limits and continuity involve measuring the distance between two real numbers x and y. This is done by calculating the absolute value of the difference between the numbers, | x − y | .