Real analysis/Introduction

The subject of real analysis is concerned with the real numbers, $ℝ$, and associated constructions. Before we can start proving theorems about the real numbers, we need to be clear on exactly what we are talking about.

We already have a fairly good idea of what numbers are like, but it is difficult to pin down what exactly we mean by a real number. Is $\sqrt{-1}$ a real number? Is $\frac{1}{0}$? It turns out that they are not (the first is a complex number and the second is undefined) but we need to be able to explain why this is.

The approach we take to set this matter straight is to define the real numbers axiomatically. In layman's terms, we set down the properties which we think the real numbers ought to have. We then prove from these properties and these properties only that the real numbers behave in the way which have come to understand that numbers behave.

Note: A table of the math symbols used below and their definitions is available here.