What exactly is a sequence? (Construction of reals) 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!
 A: 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.
A: 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.
A: 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.
A: 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.
