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
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:


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.


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:
  • ordering means that for any two cuts $\{A|B$\}$ and $\{C|D\}$, either $A\subset C$ (A is less than C), $C\subset A$ (A is greater than C), or A=C (A is equal to C);
  • field means that it is possible to define addition and multiplication, and they satisfy the desired properties;
  • complete means that any set in $\mathbb{R}$ has a least upper bound, that is an element which is less than all other upper bounds, yet greater than any element in the set.

Essentially, completeness means that there are no "holes" or "gaps" in the real number line. (The rationals $\mathbb{Q}$ are an example of an incomplete set, with lots of holes.)

Maximums, Upper Bounds, and Supremums

Given a subset $S\subset\mathbb{R}$ of the real numbers, does it have a maximum value? Consider the case where S is the set of negative real numbers demonstrates. In particular, there is no negative real number which is closest to 0 since if $r\in\mathbb{R}$, then $\frac{r}{2}<0$ and is closer to 0 then r. So the maximum of a set may not exist. But our intuition still tells us that "something like" a maximum may exist… in particular, although 0 is not a negative number, it is "the first number larger than all negative numbers". This leads to the definition of the supremum of a subset of real numbers.


The supremum of a subset $S\subset\mathbb{R}$ is $\sup(S)=\min\{x\in\mathbb{R}:x\geq s\text{ for all }s\in S\}$.

A more fitting term for this is perhaps least upper bound, since out of all the numbers greater than everything in S (that is, all the upper bounds), $\sup(S)$ is the smallest. For example, the open interval $(-3,\sqrt{2})$ has supremum $\sqrt{2}$, as does the closed interval $[-3,\sqrt{2}]$. The supremum of the set of negative numbers is zero.

It is important to note that the supremum of the set may not be contained in the set. If it is, then the number is also the maximum of the set. Perhaps an even greater question is whether the least upper bound actually exists. Since we've decided that the minimum of a set may not always be defined, one must be careful to show that the supremum actually exists for subsets of the real numbers. This is the least upper bound property of the real numbers, which will show up later as the "completeness" of the real numbers.

Theorem (Least Upper Bound Property)

For any subset $S\subset\mathbb{R}$, the set $\{x\in\mathbb{R}:x\geq s\text{ for all }s\in S\}$ has a minimum value in the set.

This discussion generalizes to the infimum or greatest lower bound of a subset $S\subset\mathbb{R}$. Note that if $S\subset\mathbb{Q}$, then $\sup(S)$ may not exist in $\mathbb{Q}$, that is, $\{x\in\mathbb{Q}:x\geq s\text{ for all }s\in S\}$ may not have a minimum value. A typical example is the set of rational numbers whose square is less than 2. The set of upper bounds is the set of rationals whose square is greater than 2, and has no minimum. So the rational numbers do not satisfy the Least Upper Bound property.

Sequences of Real Numbers

A sequence of real numbers is an infinite list of real numbers $a_1, a_2, a_3, \ldots, a_k, \ldots$. In shorthand, the sequence is often denoted $(a_n)$. It turns out that many properties of the real numbers can be conveniently characterized in terms of sequences, and their limits. A limit is a real value that the a sequence is getting arbitrarily close to. In precise terms


The limit of a sequence $(a_n)$ is said to exist and is equal to x if for every $\epsilon>0$ there exists a natural number $N\in\mathbb{N}$ such that for all $n>N$, the distance between an and x is less than $\epsilon$, that is $|a_n-x|<\epsilon$. If there is no x for which this is true, the limit does not exist. Alternately, if the limit exists the sequence converges, while if the limit does not exist the sequence diverges.



Going Further

The Road Ahead

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