Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

I am working through an Analysis textbook and came to the construction of the reals using Cauchy Sequences. I understood the proof more or less but far from completely / intuitively.

I have no image what exactly a sequence is.. does this construction mean we can have a special sequence to represent each real number we want? If so, how would a sequence for let's say $ \sqrt2 $ look like and what is the function creating this sequence?

I would be glad to get any information which could help clear this up. Of if you have any good intuition to share :)

Thank you!

share|cite|improve this question
up vote 2 down vote accepted

Here's one sequence for $\sqrt 2$:


Here's a different sequence for $\sqrt 2$:

$$ 1\\ 1.5\\ 1.4\\ 1.416666\ldots\\ 1.41379310344827586206\ldots\\ 1.4142857142857\ldots\\ \vdots $$

(Here the elements of the sequence are $\frac11, \frac32, \frac75, \frac{17}{12},\ldots$, where each fraction $\frac ab$ is followed by $a+2b\over a+b$.)

Each real number has its own sequences that are different from the sequences that other real numbers have. But each real number has many sequences that converge to it.

share|cite|improve this answer
Amazing how fast a question gets answered here! May I ask how did you come up with $ \frac{a+2b}{a+b} $ ? – want_to_know May 25 '12 at 13:13
Sorry, found the algorithm here: – want_to_know May 25 '12 at 13:18
There are several answers to that. I happen to know that one off the top of my head, because it comes up a lot, both here and elsewhere. It's an exercise in the little Rudin book Principles of Mathematical Analysis, which is where I met it first. It pops up if you just try to estimate $\sqrt 2$ by tabulating $n^2$ and $2n^2$ and looking for numbers in the two columns that are close together. It is ancient knowledge. See Pell numbers for many details. – MJD May 25 '12 at 13:20
Thanks a lot for the advice! – want_to_know May 25 '12 at 13:24
Strangely, someone asked here about that Rudin exercise only a few minutes later. (…) As I said, it comes up a lot! – MJD May 25 '12 at 13:42

Any function $\,f:\mathbb{N}\to\mathbb{Q}\,$ is a rational sequence, where we usually denote $\,a_1:=f(1)\,,\,a_2:=f(2)\,,...\,$ . The same can be done with

the reals or complex instead of the rationas.

As you talk of construction of the reals by means of Cauchy sequences I focused first at rational sequences.

Added The construction I know for the reals by means of rationa Cauchy seq's is as

follows: first, define $\,\displaystyle{R:=\left\{\{a_n\}\subset \mathbb{Q}\,/\,\{a_n\} \text{ is Cauchy}\right\}}\,$ , and define on this set the "usual"

operations of addition and multiplication coordinatewise. Then, $\,R\,$ becomes a unitary

commutative ring and $\,\displaystyle{M:=\left\{\{a_n\}\in R\,/\,\lim_{n\to\infty}a_n=0\right\}}\,$ is a maximal ideal in it, thus

$\,R/M\,$ is a field...yes, the field of real numbers.

Of course, there are several things to prove there but this is the idea.

share|cite|improve this answer
Interesting explanation! I will dig deeper into that! – want_to_know May 25 '12 at 13:25

A sequence is an infinite list of numbers (in our case rational numbers), indexed by the positive integers. We say that a sequence is Cauchy if it has a certain property which assures that the elements are getting closer and closer to each other.

You can consider $\sqrt 2$ in its decimal expansion, and then the sequence would be:

$$1, 1.4, 1.41,\ldots$$

Any other base and any other real number can work too.

share|cite|improve this answer
just to add that this sequence is Cauchy: if $(x_n)_{n\geq 1}$ is an expansion of a real number with a base $10$ as in example you have $|x_n - x_m|\leq 10x_1\cdot 10^{-|m-n|}$ – Ilya May 25 '12 at 13:08

The point of the construction by equivalence classes of Cauchy Sequences is that there is no special sequence for a given real number. As Asaf points out, there are some ways of picking out a special sequence, but the construction does not require these sequences to be picked out a priori.

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.