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!

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.
