In the book I am reading about Functional analysis, there is an algorithm for finding if some space is Banach.
1) search for element (x) in the space that we suspect to be the limit of ($x_n$)
2) check that x is element of X ....
For example we want to show that C[0,1] is complete equipped with supremum norm. The proof goes like this: Let $x_n$ be Cauchy sequence in C[0,1] ... then we find x and show that x is indeed continuous function.
My question is - Why we can choose(find) a sequence ($x_n$) in the space. How do we know it exist for sure. How are we sure that there exists sequences in more abstract spaces(other than $R, R^n$)?