It is currently Tue Mar 26, 2019 3:24 pm


All times are UTC [ DST ]




Post new topic Reply to topic  [ 2 posts ] 
Author Message
PostPosted: Wed Mar 09, 2016 6:12 pm 
Team Member

Joined: Mon Nov 09, 2015 1:52 pm
Posts: 426
Let \(\displaystyle{\left(X,d\right)}\) be a metric space and \(\displaystyle{D}\) a dense subset of \(\displaystyle{X}\) having the

property : Each \(\displaystyle{\rm{Cauchy}}\) sequence of elements of \(\displaystyle{D}\) converges to \(\displaystyle{X}\) .

Prove that the metric space \(\displaystyle{\left(X,d\right)}\) is complete.



Comment : The metric space \(\displaystyle{\left(\mathbb{R},|\cdot|\right)}\) with \(\displaystyle{D=\mathbb{Q}}\) is an example of such space.


Top
Offline Profile  
Reply with quote  

PostPosted: Wed Mar 09, 2016 6:13 pm 
Administrator
Administrator
User avatar

Joined: Sat Nov 07, 2015 6:12 pm
Posts: 841
Location: Larisa
Papapetros Vaggelis wrote:
Let \(\displaystyle{\left(X,d\right)}\) be a metric space and \(\displaystyle{D}\) a dense subset of \(\displaystyle{X}\) having the

property : Each \(\displaystyle{\rm{Cauchy}}\) sequence of elements of \(\displaystyle{D}\) converges to \(\displaystyle{X}\) .

Prove that the metric space \(\displaystyle{\left(X,d\right)}\) is complete.



Comment : The metric space \(\displaystyle{\left(\mathbb{R},|\cdot|\right)}\) with \(\displaystyle{D=\mathbb{Q}}\) is an example of such space.


Pick a Cauchy sequence say \( x_n \). Since the set \(\displaystyle{D}\) is dense there exist for every \( n \) an element \( y_n \in D \) such that \( d(x_n, y_n)<\frac{1}{n} \). Using the relation:

$$d(y_m, y_n)\leq d(y_m, x_m) + d(x_m, x_n) + d(x_n, y_n)$$

we can easily see that \( y_n \) is Cauchy. Hence converges. Let \( x \in X\) be its limit. Since:

$$0\leq d(x_n, y) \leq d(x_n, y_n) +d(y_n, y)$$

we easily see that \( x_n \) converges to \( y \), completing the proof.

_________________
Imagination is much more important than knowledge.
Image


Top
Offline Profile  
Reply with quote  

Display posts from previous:  Sort by  
Post new topic Reply to topic  [ 2 posts ] 

All times are UTC [ DST ]


Mathimatikoi Online

Users browsing this forum: No registered users and 1 guest


You cannot post new topics in this forum
You cannot reply to topics in this forum
You cannot edit your posts in this forum
You cannot delete your posts in this forum
You cannot post attachments in this forum

Search for:
Jump to:  
cron
Powered by phpBB® Forum Software © phpBB Group Color scheme created with Colorize It.
Theme created StylerBB.net