Analysis Day 1: Few Questions about Cauchy sequences

Question

Sorry for asking these questions; in the textbook I'm using, there are instances where I just don't get how what the author said in paragraph B follows from paragraph A.

1. Why do we need to use Cauchy sequences to define the real numbers? Can't we just define a real number $x$ as a number that all sequences converge to if there is a convergence? There doesn't seem to be a difference.

2. I know that a Cauchy sequence has a real number as a limit. But how do we know that every real number is the limit of a Cauchy sequence of rational numbers? Is it because every real number has an infinite decimal expansion, and we can use each term to be the term in a Cauchy sequence?

On a more meta level, however, I'm confused because we are trying to create a decimal expansion for a number that we are trying to show exists but the only way to create a decimal expansion is to already have the number in existence.

Answer 1

You do not need Cauchy sequences to define real numbers from rationals. There are other procedures.

The whole point is that you are constructing reals from rationals so the construction cannot contain a reference to the (yet undefined) real number as you propose. You cannot say that $\sqrt{2}$ is the set of sequences that converge to $\sqrt{2}$ . You need an internal definition. In Cantor’s construction equivalent Cauchy sequences are used to that end.

Answer 2

There are several ways to build real numbers. I imagine that you refer here to the one that uses Cauchy sequences of rational numbers. Regarding your questions.

Why do we need to use Cauchy sequences to define the real numbers? Can't we just define a real number $x$ as a number that all sequences converge to if there is a convergence? There doesn't seem to be a difference.

The problem is that for the construction of the reals, we're using sequences of rationals... as the reals are not yet defined! And how can you say that a sequence of rationals converges to a number that doesn't exist as not yet defined!!!

I know that a Cauchy sequence has a real number as a limit. But how do we know that every real number is the limit of a Cauchy sequence of rational numbers? Is it because every real number has an infinite decimal expansion, and we can use each term to be the term in a Cauchy sequence?

This is where the definition of real numbers comes into play. A real number is defined as an equivalence class of Cauchy sequences of rational numbers. Two Cauchy sequences are called equivalent if and only if the difference between them tends to zero. Based on this construction, every Cauchy sequence of rationals defines a unique real number.

Answer 3

In this construction, a real number is not defined as the limit of a Cauchy sequence of rationals. Indeed, this definition wouldn't make sense. Since irrational numbers haven't been defined yet, we have to work only with the rational numbers. This presents quite a few technical problems: in the rational numbers, the sequence $(1,1.4,1.41,1.414,\dots)$ does not converge! Therefore, it makes no sense to define $\sqrt{2}$ as the "limit" of this sequence. Although the above sequence does converge in the real numbers, the real numbers haven't been constructed yet!

These problems lead to a definition of "real number" that is as clever as it is counter-intuitive. We might consider actually defining $\sqrt{2}$ as the sequence $(1,1.4,1.41,1.412,\dots)$.* On the surface, this definition seems absurd. Why should a real number—something that we intuitively think of as a point on the number line—be a sequence? To answer this question, we need to draw a distinction between how we think about real numbers in "everyday" mathematics, and how they are formally constructed in set theory. It helps to consider how number systems are constructed in general.

It is common to construct the natural numbers as sets. We start by defining $0$ as the empty set $\{\}$, and $1$ as $\{0\}$, and $2$ as $\{0,1\}$, and so on. Does this mean that when you ask a mathematician to think of the number $1$, she pictures the set containing the empty set? Of course not! This definition is accepted in mathematics not because it aligns with how we intuitively think of the natural numbers, but because it works—in other words, because when we follow this construction, we end up with a number system that behaves in the way we expect it to. We end up with a number system that has the same properties as the everyday natural numbers that we are introduced to when we are children. The fact that we had to choose a seemingly unnatural construction is irrelevant.

The next question that we must ask ourselves is why the definitions $0=\{\},1=\{0\},2=\{0,1\}$, etc. do end up producing a number system that aligns with our intuition. Here is one possible answer. We know, from our everyday conception of the natural numbers, that each natural number is completely determined by the natural numbers that come before it. If I tell you that the only numbers preceding a certain natural number are $0,1,2,3,4,$ and $5$, then you know that that natural number must be $6$. This means that any statement about the natural number $6$ can be seen as an abbreviation of a statement about the set $\{0,1,2,3,4,5\}$. Since our intuition tells us that the set $\{0,1,2,3,4,5\}$ seems to completely determine the number $6$, it does seem so unreasonable to actually define $6$ as the set $\{0,1,2,3,4,5\}$ in set theory. The beauty of this formal definition is that it makes no reference to the number $6$ at all, avoiding all circularity. Again, I am not claiming that this is how we think of the number $6$ in practice. The reason that we are conducting this process is to convince ourselves that the everyday number systems we work with do not lead to any sort of contradiction. Once we have finished this process, we can go back to thinking of numbers as objects to count with, or as points on a number line.

Once we have constructed the natural numbers, we can construct the integers, and then the rational numbers. By this point, we have convinced ourselves that the rational numbers can be defined formally. Therefore, there is no harm in thinking of a rational number as a number of the form $p/q$, where $p$ and $q$ are integers, and $q\neq0$. The fact that rational numbers are themselves sets in set theory is not relevant to us now. Our proposed definition of real number is that is a certain sequence of rational numbers. For instance, $\sqrt{2}$ is defined as $(1,1.4,1.41,1.414,\dots)$. Unfortunately, this definition is a little unsatisfying. We know, from our everyday conception of real numbers, that there are many sequences that converge to $\sqrt{2}$, and it is not clear why we should prefer one sequence over another. This becomes particularly evident when we realise that, ultimately, the only reason we favour the sequence $(1,1.4,1.41,1.414,1.4142\dots)$ is because we have $10$ fingers on our hands! It is this observation that leads to the following definition:

A real number is an equivalence class of Cauchy sequences of rationals.

The definition of "equivalence class" takes a little work to unpack. First of all, an equivalence relation on a set $X$ is a binary relation $\sim$ satisfying three properties:

• $a \sim a$ for all $a\in X$ (reflexivity).
• $a \sim b \iff b\sim a$ for all $a,b\in X$ (symmetry).
• $a\sim b \land b\sim c\implies a \sim c$ for all $a,b,c\in X$ (transitivity).

In our case, $X$ equals the set of all Cauchy sequences of rationals. We say that $(a_n)\sim(b_n)$ if $\lim_{n\to\infty}a_n-b_n=0$. Intuitively, two sequences are equivalent if they converge to the same real number. Once we have shown that $\sim$ satisfies the properties above, we can proceed to the next stage.

If $X$ is an arbitrary set, and $\sim$ is an equivalence relation defined on $X$, then we define the equivalence class of $a\in X$ as the set $$ [a]=\{x\in X:x\sim a\} \, . $$ Here, $X$ and $\sim$ are defined as before, and so the equivalence class of a sequence $(a_n)$ is the set $$ [(a_n)]=\left\{(b_n)\in X:\lim_{n\to\infty}a_n-b_n=0\right\} \, . $$ From this point, we have defined what a real number is. Still, this leaves upon many questions. For instance, how do we know that every infinite decimal is a real number, and conversely, that every real number can be written as a decimal? How do we define addition and multiplication of real numbers, which, in set theory, are different operations to addition and multiplication of rational numbers? Is the rational number $1$ literally the same as the real number $1$?

All of these questions can be answered, but, for that, I would suggest turning to an Analysis book, rather than Mathematics Stack Exchange. Terence Tao's Analysis I answers all of these questions in much greater than detail than I can in one post. Still, I hope my answer explains the philosophy behind how we construct the real numbers, and that this will give you some footing for your future studies.

---

*The $n$-th term of this sequence can be defined as $$ a_n=\frac{m}{10^n} \, , $$ where $m$ is the largest integer such that $(a_n)^2<2$.

Answer 4

Let's say we've constructed the rationals $\mathbb Q$ to be (equivalence classes of) pairs of integers $(p,q)\simeq\frac{p}{q}$ with $q\neq 0$. We know how to add, subtract, multiply, and divide them, and life is good.

Next we start talking about sequences of rational numbers. We call a sequence $\{x_n\}$ Cauchy if, for any $\epsilon>0$, there exists some $N\in \mathbb N$ such that $n,m>N \implies |x_n-x_m|<\epsilon$. We say a sequence $\{x_n\}$ converges to a limit $x$ if, for any $\epsilon>0$, there exists some $N\in\mathbb N$ such that $n>N\implies |x_n-x|<\epsilon$.

These notions are distinct. In a Cauchy sequence, the terms in the sequence eventually get (and remain) arbitrarily close to one another; in a convergent sequence, the terms in the sequence eventually get (and remain) arbitrarily close to some fixed "target" $x\in \mathbb Q$.

It's easy to show that all convergent sequences are Cauchy, but the converse is generally not true; there are Cauchy sequences in $\mathbb Q$ which do not converge. For example, consider the sequence of rationals $x_n = p_n/q_n$, where $p_n$ is the concatanation of all of the natural numbers $\leq n$ (e.g. $p_5=12345$) and $q_n = 10^n$. A few select terms from this sequence are $$x_1 = 0.1 \qquad x_3 = \frac{123}{10^3}=0.123 \qquad x_{12} = \frac{123456789101112}{10^{12}}=0.123456789101112$$ It's not difficult to show that this sequence is Cauchy. However, it's also not too difficult to see that it does not converge, since every rational number has a terminating or repeating decimal expansion and the expansion being built up term-by-term in this sequence is clearly neither of these things.

So, we're faced with a problem. Cauchy sequences intuitively exhibit all the signs of convergence, but for some of them there's no target $x\in \mathbb Q$ at the metaphorical end of the rainbow. You might intuitively describe this as being due to the fact that $\mathbb Q$ has "holes"; more technically, we call such spaces topologically incomplete.

---

If we want to patch the holes, we simply define an equivalence relation $\sim$ between Cauchy sequences, where $\{x_n\}\sim\{y_n\} \iff \{x_n-y_n\}$ converges to $0$, and define a new set of numbers $\mathbb R$ to be the corresponding equivalence classes $[x_n]$. In this way, $\mathbb R$ is topologically complete by construction (it is, in fact, called the topological completion of $\mathbb Q$).