Metric space and dense subset
-
- Community Team
- Posts: 426
- Joined: Mon Nov 09, 2015 1:52 pm
Metric space and dense subset
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.
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.
- Tolaso J Kos
- Administrator
- Posts: 867
- Joined: Sat Nov 07, 2015 6:12 pm
- Location: Larisa
- Contact:
Re: Metric space and dense subset
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: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.
$$d(y_m, y_n)\leq d(y_m, x_m) + d(x_m, x_n) + d(x_n, y_n)$$
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.
Create an account or sign in to join the discussion
You need to be a member in order to post a reply
Create an account
Not a member? register to join our community
Members can start their own topics & subscribe to topics
It’s free and only takes a minute
Sign in
Who is online
Users browsing this forum: No registered users and 21 guests