I have started studying Functional Analysis following "Introduction to Functional Analysis with Applications". In chapter 1-6 there is the following proof
For any metric space $X$, there is a complete metric space $\hat{X}$ which has a subspace $W$ that is isometric with $X$ and is dense in $\hat{X}$
(Page 1 & 2) http://i.imgur.com/CRXjh.png
(Page 3 & 4) http://i.imgur.com/PogqC.png
I think I understand parts (a) and (b). At the top of page 3, section (c) where it is proving $\hat{X}$ is complete it states:
Let $(\hat{x_{n}})$ be any Cauchy Sequence in $\hat{X}$. Since $W$ is dense in $\hat{X}$, for every $\hat{x_{n}}$, there is a $\hat{z_{n}}\varepsilon W$ such that $\hat{d}(\hat{x_{n}},\hat{z_{n}}) < \frac{1}{n}$
I do not understand why we choose $ \frac{1}{n}$. Would some ε > 0, for each $\hat{x_{n}}$, not suffice? I assume it must not, but I don't see why, so I must not understand this proof. Maybe i'm not sure on what n is referring to because of the subscripts n on the lefthand side. Is n the index of the Cauchy sequence in $\hat{X}$, $(\hat{x_{n}})$? Is it the index of the Cauchy sequence in X, $\hat{x_{n}}$?
Any help would be greatly appreciated, i am pretty dumb and this has puzzled me for a couple days.