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


by Petrarch
Tags: completion, metric, proof, space
Petrarch
Petrarch is offline
#1
Jul1-12, 12:48 PM
P: 2
1. The problem statement, all variables and given/known data

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 [itex]X[/itex], there is a complete metric space [itex]\hat{X}[/itex] which has a subspace [itex]W[/itex] that is isometric with [itex]X[/itex] and is dense in [itex]\hat{X}[/itex]

(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 [itex]\hat{X}[/itex] is complete it states:

Let [itex](\hat{x_{n}})[/itex] be any Cauchy Sequence in [itex]\hat{X}[/itex]. Since [itex]W[/itex] is dense in [itex]\hat{X}[/itex], for every [itex]\hat{x_{n}}[/itex], there is a [itex]\hat{z_{n}}\varepsilon W[/itex] such that [itex]\hat{d}(\hat{x_{n}},\hat{z_{n}}) < \frac{1}{n}[/itex]

I do not understand why we choose [itex] \frac{1}{n}[/itex], 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.
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