Real Analysis I
 Level: 0 1 2 3 4 5 6 7
MSC Classification: 26 (Real functions)

Prerequisites: Set Theory, Calculus I

# Getting Oriented

 Rough Guides to Analysis
 Calculus Real Analysis Complex Analysis

Real analysis is the study of the properties of real numbers, and functions on the real numbers. It demonstrates how to obtain the results usually seen in a calculus course using a completely rigorous mathematical approach. Frequently, one sees a lot of new and unfamiliar notation in real analysis, as well as new techniques for proving things. For this reason, real analysis has a reputation as being a rather difficult course. It is also full of surprises, and results which run counter to intuition. For example, there are more irrational numbers than rational numbers, even though both sets are infinite.

In the big picture, much of analysis involves getting at ideas which are rather intuitive in a mathematically rigorous way. For example, the definition of continuity is very intuitive. Informally, it can be thought of as being able to trace a function on a piece of paper without lifting up the pencil. But the mathematical definition is not so easy, involving sets, magnitudes, and Greek letters. So a rather large part of analysis is becoming familiar enough with the notation to be able to see the more intuitive concept amidst all the notation.

# The Basics

## Defining the Real Numbers

Usually, the real numbers are thought of as numbers on a line or as decimal representations. But there is also a way they can be constructed directly from the set of integer fractions $\frac{p}{q}$ (the rational numbers $\mathbb{Q}$). The advantage to this approach is that the rationals $\mathbb{Q}$ can be constructed from the integers $\mathbb{Z}$, which can in turn be constructed from the natural numbers $\mathbb{N}$. And using this approach one can prove all the properties of real numbers that ought to be true, such as the existence of multiplication, addition, division, and ordering.

Given only the rational numbers, is there any way to get at the concept of $\sqrt{2}$, which is not a rational number? You can't get at it directly, but you could talk about all the rational numbers whose square is less than 2, and all the rational numbers whose square is greater than 2. This idea, splitting the rational numbers into two distinct pieces, leads to the following definition:

Definition

A cut is a partition of the rational numbers into two disjoint subsets A and B such that all elements of A are less than all elements of B, and A contains no largest elements. It is denoted $A|B$.

For example, if A consists of all the rational numbers less than $\sqrt{2}$ and B of everything else, then $A|B$ is a cut. The beauty of this approach is that every real number corresponds to a unique cut, and all properties of the reals can be proven on the level of cuts.

Theorem

The set of cuts $\{A|B\}$ is in a 1:1 correspondence with the real numbers $\mathbb{R}$, and satisfies the (i) ordering, (ii) field, and (iii) completeness properties of real numbers.

The meaning of these three properties is: