It is currently Thu Jun 20, 2019 12:43 pm

 All times are UTC [ DST ]

 Page 1 of 1 [ 2 posts ]
 Print view Previous topic | Next topic
Author Message
 Post subject: Metric space and dense subsetPosted: 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

 Post subject: Re: Metric space and dense subsetPosted: Wed Mar 09, 2016 6:13 pm

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.

Top

 Display posts from previous: All posts1 day7 days2 weeks1 month3 months6 months1 year Sort by AuthorPost timeSubject AscendingDescending
 Page 1 of 1 [ 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 forumYou cannot reply to topics in this forumYou cannot edit your posts in this forumYou cannot delete your posts in this forumYou cannot post attachments in this forum

Search for:
 Jump to:  Select a forum ------------------ Algebra    Linear Algebra    Algebraic Structures    Homological Algebra Analysis    Real Analysis    Complex Analysis    Calculus    Multivariate Calculus    Functional Analysis    Measure and Integration Theory Geometry    Euclidean Geometry    Analytic Geometry    Projective Geometry, Solid Geometry    Differential Geometry Topology    General Topology    Algebraic Topology Category theory Algebraic Geometry Number theory Differential Equations    ODE    PDE Probability & Statistics Combinatorics General Mathematics Foundation Competitions Archives LaTeX    LaTeX & Mathjax    LaTeX code testings Meta
The text passage contains a mathematical question and a solution to that question. The question is about proving that a metric space is complete given certain conditions about a dense subset. The solution constructs a Cauchy sequence in the dense subset and shows that the original sequence converges, thus proving the metric space is complete.
single_question_with_answer
Given a text passage, determine:
    1. If it contains a mathematical question (including word problems or calculations)
    2. If an answer to that mathematical question is provided in the text

First, describe the content structure you found in 1-2 sentences.

Then provide your analysis in the following format:
TYPE: <one of the following values>
- NO_QUESTION
- SINGLE_QUESTION
- SINGLE_QUESTION_WITH_ANSWER
- MULTIPLE_QUESTIONS
- MULTIPLE_QUESTIONS_WITH_ANSWERS
system
It is currently Thu Jun 20, 2019 12:43 pm

 All times are UTC [ DST ]

 Page 1 of 1 [ 2 posts ]
 Print view Previous topic | Next topic
Author Message
 Post subject: Metric space and dense subsetPosted: 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

 Post subject: Re: Metric space and dense subsetPosted: Wed Mar 09, 2016 6:13 pm

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.

Top

 Display posts from previous: All posts1 day7 days2 weeks1 month3 months6 months1 year Sort by AuthorPost timeSubject AscendingDescending
 Page 1 of 1 [ 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 forumYou cannot reply to topics in this forumYou cannot edit your posts in this forumYou cannot delete your posts in this forumYou cannot post attachments in this forum

Search for:
 Jump to:  Select a forum ------------------ Algebra    Linear Algebra    Algebraic Structures    Homological Algebra Analysis    Real Analysis    Complex Analysis    Calculus    Multivariate Calculus    Functional Analysis    Measure and Integration Theory Geometry    Euclidean Geometry    Analytic Geometry    Projective Geometry, Solid Geometry    Differential Geometry Topology    General Topology    Algebraic Topology Category theory Algebraic Geometry Number theory Differential Equations    ODE    PDE Probability & Statistics Combinatorics General Mathematics Foundation Competitions Archives LaTeX    LaTeX & Mathjax    LaTeX code testings Meta
user
The text passage contains a mathematical question and a solution to that question. The question is about proving that a metric space is complete given certain conditions about a dense subset. The solution constructs a Cauchy sequence in the dense subset and shows that the original sequence converges, thus proving the metric space is complete.

TYPE: SINGLE_QUESTION_WITH_ANSWER