Lecture 22

It’s possible to prove that a sequence is Cauchy directly from the definition. Here’s the simplest possible example:

Proposition

A constant sequence is Cauchy.

Proof

Let $a_0, a_1, a_2, \ldots$ be a constant sequence with value $a$; that is, $a_n = a$ for all $n$.

We must show that, for any $\epsilon>0$, there is an $N$ such that, for all $m,n>N$, we have $\left|a_m-a_n\right|<\epsilon$.

In fact, no matter what $\epsilon$ is, we can choose $N=0$, because for any $m$ and $n$ whatsoever we have $\left|a_m-a_n\right| = \left|a-a\right| = \left|0\right| = 0 < \epsilon,$ so the proof is done.

Of course, it’s very unusual to be able to choose one $N$ that works for every $\epsilon$.

In fact, more is true:

Theorem

Any convergent sequence is Cauchy.

Proof

Suppose we have a sequence $a_0,a_1,\ldots$ converging to $x$. We must show that it is Cauchy.

So suppose we’re given some $\epsilon>0$: we must find some $N$ such that, for all $m,n>N$ we have $\left|a_m - a_n\right|<\epsilon.$

Since it is convergent, there is an $N$ such that for all $n>N$ we have $\left|a_n - x\right|<\frac{\epsilon}{2}.$

We’ll use that $N$; because then we have $\begin{aligned} {\left|a_m - a_n\right|}& {\leq\left|a_m-x\right|+\left|a_n-x\right|\qquad\text{(by the triangle inequality)}}\\ &{<\frac{\epsilon}{2}+\frac{\epsilon}{2}}\\ &{=\epsilon,}\end{aligned}$ exactly as required.

How to construct $\mathbb{R}$

Now we have the tools to understand what the reals really are (no pun intended).

Let’s give it in context. What follows is revisionist history: things didn’t actually happen exactly like this, but maybe they should have done.

• In the beginning there was $\mathbb{N}$.

  We’ve said only a little about constructing the naturals, but we could have said more.

• $\mathbb{Z}$ was invented from $\mathbb{N}$ by insisting that one should be able to subtract.

  In other words, new numbers were invented, in order to be the values obtained by previously impossible subtractions in the naturals. So we invented $-2$ to be $0-2$ and $-137$ to be $0-137$.

  You don’t want one new number for each subtraction. For example, we want to have $5-7=-2$ and $1000000-1000002=-2$, as well.

  But that’s okay: the theory is workable, and we get $\mathbb{Z}$ by doing it. Nobody gets confused about which integers are equal. We can make definitions like $a-b = c-d\qquad\text{if and only if $a+d=b+c$}.$

• $\mathbb{Q}$ was invented from $\mathbb{Z}$ by insisting that one should be able to divide (by things that aren’t zero).

  In other words, new numbers were invented, in order to be the values obtained by previously impossible divisions in the integers.

  So we invented $1/5$ and $-3/7$ accordingly.

  Again, we don’t want one new number for each division. We also have $100/500 = -2/-10 = 1/5$. But that’s okay as well, and we don’t get confused. We can make definitions like $a/b = c/d\qquad\text{if and only if $ad=bc$}.$

• $\mathbb{R}$ was invented from $\mathbb{Q}$ by insisting that all Cauchy sequences of rationals should converge.

  In other words, new numbers were invented, in order to be the limits of Cauchy sequences of rationals which don’t converge to a rational.

  So, for example, we get $\pi$ as the limit of the Cauchy sequence of rationals $\begin{aligned} &3,& &\frac{31}{10},& &\frac{314}{100},& &\frac{3141}{1000},& &\frac{31415}{10000},& &\ldots. \end{aligned}$

  Again, we don’t actually want one new number for each Cauchy sequence. There are other Cauchy sequences of rationals that converge to $\pi$ (some of them more interesting, perhaps). A famous example is due to Gregory and Leibniz: $4,\quad 4-\frac{4}{3},\quad 4-\frac{4}{3}+\frac{4}{5},\quad 4-\frac{4}{3}+\frac{4}{5}-\frac{4}{7},\quad\ldots.$

One minor issue to be clear about is which numbers we take in the definition of a Cauchy sequence of rationals.

We demand that something be true for “every positive $\epsilon$”. That had better mean “every positive rational $\epsilon$” (I’d been being vague about what $\epsilon$ was). In fact it won’t make any difference in meaning.

But then, given that, we could just say that the reals are the Cauchy sequences of rationals, subject to some restriction about which ones are the same.

We need to know how to say that two Cauchy sequences “are trying to converge to the same number”, in order to say when they’re describing the same real number.

There are a number of different definitions we could use, all providing the same results. Perhaps the clearest is that two Cauchy sequences $a_0,a_1,\ldots$ and $b_0,b_1,\ldots$ of rationals converge to the same real number if, for all rational $\epsilon>0$, there is some $N$ such that for all $m,n>N$ we have $\left|a_m-b_n\right| < \epsilon.$ We might regard this as saying “no matter what is meant by close, the two sequences get close to each other and stay close to each other forever”.

This seems a very sensible way of describing the real numbers: it says that they fill in the gaps in the rationals, the things that sequences might try to converge to.

In particular, I hope you agree that it’s a more natural way of understanding the reals than talking about decimal expansions.

If you insist on working with decimals, then you can do so happily with Cauchy sequences of rationals. For example, suppose that you insist on talking about the decimal $1.414213562373\cdots$ (This happens to be the decimal expansion of the square root of $2$.)

Then this can be accommodated in our construction easily, using the trick I mentioned earlier: we can represent it as the limit of the Cauchy sequence $1,\quad \frac{14}{10},\quad \frac{141}{100},\quad \frac{1414}{1000},\quad \frac{14142}{10000},\quad\ldots.$

But that Cauchy sequence doesn’t seem too exciting. There are others, and some of them tell us more about what the square root of $2$ really is.

For example, there’s the Newton iteration scheme. The details of this are not really part of this course, but it tells us that if $x$ is an approximation to $\sqrt{2}$, then $\frac{x}{2}+\frac{1}{x}$ is a better approximation.

If we start with $1$ as an approximation, then this gives us the sequence $1,\quad \frac{3}{2},\quad \frac{19}{12},\quad \frac{577}{408}, \quad \frac{665857}{470832}, \ldots$ It’s not hard to imagine that this is a much better way of describing $\sqrt{2}$ than its decimal expansion: easier to prove things about it than some weird string of digits.

Decimal expansions are, of course, still useful for dealing with approximate forms of reals. But it’s nice to have alternatives, and extremely useful to have a system that doesn’t depend on them.

Quite a lot of pre-20th century mathematics can be regarded as giving interesting facts about Cauchy sequences of rationals, either in general or in specific instances. These facts I’ve given you for $\pi$ and $\sqrt{2}$ are just two parts of a very rich tapestry!

What’s next?

If you liked some of the topics in this course, may be wondering what’s next.

So far as number theory goes, our elementary methods will give out sooner or later. A good next step is to learn lots of algebra. (There are other good reasons to do that.) Next semester, Evgeny will take this up. You can return later in your degrees to a huge range of questions of which equations are solvable in which systems of numbers.

The way forward with real analysis is much more straightforward.

From where you are, it’s easy to define the derivative formally and start proving things about it. You’re probably not scared of differentiation, but you perhaps should be cautious of differentiating functions of several variables.

Integration is a bit more technical, and will require more work. There are great rewards there, though: the theory of integration is vital to the foundations of probability theory.

Sooner or later, you can try using the same techniques in more exotic surroundings: the concepts of approximation we started talking about in this course give us a way in to studying abstract concepts of space.