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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!

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4 Answers 4

up vote 2 down vote accepted

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

$$1\\1.4\\1.41\\1.414\\1.4142\\\vdots$$

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.

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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
1  
Sorry, found the algorithm here: en.wikipedia.org/wiki/Square_root_of_2 –  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. (math.stackexchange.com/questions/149646/…) 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.

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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.

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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.

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