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!

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.
