Completion of Metric Space Proof from Intro. to Func. Analysis w/ Applications

[Petrarch]

Completion of Metric Space Proof from "Intro. to Func. Analysis w/ Applications"

Homework Statement

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 n, not suffice? I assume it must not, but I don't see why, so I must not understand this proof.

Any help would be greatly appreciated, i am pretty dumb and this has puzzled me for a couple days.

 

[lippyka]

Homework Equations Not applicable. The Attempt at a SolutionWe choose \frac{1}{n} because if we were to pick any ε > 0, it is possible that two distinct Cauchy sequences could converge to the same limit, which would violate the uniqueness of limits in a metric space. By choosing \frac{1}{n} for each n, we guarantee the uniqueness of limits in \hat{X}.