Real Analysis/Constructing the real numbers

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Real Analysis Constructing the real numbers

Exercises→

This construction will use some advanced concepts in mathematics and so it is recommended that this chapter be studied only after you are familiar with equivalence classes and embeddings. It can be safely skipped till you study these concepts.

Until now we have been following the axiomatic approach in our study of real numbers. That is, we have assumed that there exists a set of real numbers with certain axioms. However in mathematics one tries to make such assumptions as little as possible. At the bare foundations we actually make (without possibly knowing it) some assumptions regarding sets, and it would be nice if we needn't add any more. In fact using only the assumptions concerning sets it is possible to prove that a set of rationals exist. So our job is to actually construct the reals using the available rationals, in order to prove that the axioms of the real numbers are consistent and exist under ZFC.

We'll begin our construction with a plethora of definitions.

Sequences

These definitions are more thoroughly explored in the sequences section of the book. The main difference is that the elements of the sequences we describe here are rational numbers $\mathbb {Q}$ not real numbers $\mathbb {R}$ .

They are repeated here for convenience but for greater detail please refer to the appropriate section.

A sequence of rational numbers is any function $x:\mathbb {N} \to \mathbb {Q}$ and is denoted by $(x_{n})$ .

Special Sequences

A sequence of rationals $(x_{n})$ is a null sequence if for each rational $\varepsilon >0$ , there is some $N$ such that for evey $n>N$ we have $|x_{n}|<\varepsilon$ .

A rational sequence $(x_{n})$ is a constant sequence if there is some rational $x$ such that $x_{n}=x$ for every $n\in \mathbb {N}$ .

Addition and Multiplication of Sequences

Now define addition of two sequences $(f_{n})$ and $(g_{n})$ to be the sequence ${\bigl (}(f+g)_{n}{\bigr )}$ , where $(f+g)_{n}=f_{n}+g_{n}$ .

Also define the multiplication of $(f_{n})$ and $(g_{n})$ as the sequence ${\bigl (}(f\cdot g)_{n}{\bigr )}$ , where ${\bigl (}(f\cdot g)_{n}{\bigr )}=f_{n}\cdot g_{n}$ .

Negation

Let $(f_{n})$ be a rational sequence. Then the negation of $(f_{n})$ , denoted $-(f_{n})$ , is defined as the rational sequence $(-f_{n})$ , i.e., the sequence $(g_{n})$ where $g_{n}=-f_{n}$ .

Subtraction

Let $(f_{n})$ and $(g_{n})$ be two sequences of rational numbers. Then we define the difference of $(f_{n})$ and $(g_{n})$ , denoted ${\bigr (}(f-g)_{n}{\bigl )}$ , as the sequence $(f_{n})+{\bigl (}-(g_{n}){\bigr )}$ .

Cauchy Sequences

A sequence of rational numbers $(x_{n})$ is Cauchy if for each rational $\varepsilon >0$ , there exists an $N$ such that for every $m,n>N$ we have $|x_{n}-x_{m}|<\varepsilon$ .

Let ${\mathcal {C}}$ denote the set of such rational Cauchy sequences. It is left as an exercise to show that null and constant sequences defined above are Cauchy. It is an exercise to show that if $(f_{n})$ and $(g_{n})$ are Cauchy, then both ${\bigl (}(f+g)_{n}{\bigr )}$ and ${\bigl (}(f\cdot g)_{n}{\bigr )}$ are also in ${\mathcal {C}}$ . It is a simple step to further show that ${\bigl (}(f-g)_{n}{\bigr )}$ is in ${\mathcal {C}}$ .

Equivalence of Sequences

Define a relation $\sim$ on ${\mathcal {C}}$ , the set of Cauchy sequences, by $f\sim g$ if and only if $f-g$ is a null sequence.

Now it is a simple exercise to show that $\sim$ is an equivalence relation.

The Real Numbers

We will let $\mathbb {R}$ denote the set of all equivalence classes of $\sim$ . Further we let ${\bar {f}}$ denote the equivalence class of $f$ . Our goal is to show that this set satisfies all the properties we attributed to the real numbers. Since our goal is to construct the real numbers it seems reasonable to assign our proposed set the same symbol. We must now go through all of the basic axioms of the real numbers and show that they are inherent properties of this set.

Totally Ordered Field

Now if ${\bar {f}},{\bar {g}}$ are members of $\mathbb {R}$ then it can be easily checked that $+$ and $\cdot$ defined by ${\bar {f}}+{\bar {g}}={\overline {f+g}}$ and ${\bar {f}}\cdot {\bar {g}}={\overline {f\cdot g}}$ are well-defined binary operations on Cauchy sequences.

Also the order $\leq$ can be defined by letting ${\bar {f}}\leq {\bar {g}}$ if and only if there is some rational $n_{0}$ such that for all $n$ , if $n\geq n_{0}$ then $f_{n}\leq g_{n}$ . In this way we have a set whose elements we can call as reals.

Remarks

A real number is thus the equivalence class of a special kind of sequence of rationals. Clearly the rational number 1 is not the real number 1. This seemingly counter intuitive issue is resolved by considering an embedding $F$ from the rationals to the reals defined by $F(r)={\overline {f_{r}}}$ where $f_{r}(n)=r,\forall n\in \mathbb {N}$ . Under this embedding the rational 1 can be identified with the real $F(1)$ and so the rationals can be considered as a subset of the reals.

This seems rather a strange way to define reals numbers, but in fact this is mathematically quite sound. A set constructed in this fashion behaves precisely as real numbers should do intuitively and the construction of this set involves absolutely no assumptions on our part beyond those needed for $\mathbb {Q}$ .

There is another way of constructing the reals using an approach given by Richard Dedekind. The above way was given by Georg Cantor in 1872. Dedekind also published his technique in the same year. (For those interested, Dedekind's construction has been provided in the appendix).