Every closed ball in X is complete, then X is complete

  • Thread starter mahler1
  • Start date
  • #1
222
0
Homework Statement .
Prove that if every closed ball in a metric space X is a complete subspace, then X is complete.


The attempt at a solution.

Let ##\{x_n\}## be a cauchy sequence in X. Then, for ##ε=1##, ##\exists## ##n_0 \in \mathbb N##: ##\forall## ##m≥n_0##,##n≥n_0## ##d(x_m,x_n)<1##. Consider the closed ball centered at ##x_{n_0}## of radius 1. Then, the subsequence ##\{x'_n\}## such that ##x'_n=x_{n_0 + n-1}## is a subsequence of ##\{x_n\}## included in ##K(x_{n_0},1)## . By hypothesis, the ball is a complete subspace and the subsequence ##\{x'_n\}## is cauchy. But then ##\{x'_n\}## converges to some point ##x \in K(x_{n_0},1)## This implies ##\{x'_n\}## converges to the same point x in the metric space X. So the original sequence ##\{x_n\}## has a convergent subsequence that converges to the point x, this means that ##\{x_n\}## converges to x. I've proved that an arbitrary cauchy sequence in X is convergent, this implies every cauchy sequence in X is convergent, then X is a complete metric space.


Is my proof correct?
 

Answers and Replies

  • #2
Dick
Science Advisor
Homework Helper
26,260
619
Homework Statement .
Prove that if every closed ball in a metric space X is a complete subspace, then X is complete.


The attempt at a solution.

Let ##\{x_n\}## be a cauchy sequence in X. Then, for ##ε=1##, ##\exists## ##n_0 \in \mathbb N##: ##\forall## ##m≥n_0##,##n≥n_0## ##d(x_m,x_n)<1##. Consider the closed ball centered at ##x_{n_0}## of radius 1. Then, the subsequence ##\{x'_n\}## such that ##x'_n=x_{n_0 + n-1}## is a subsequence of ##\{x_n\}## included in ##K(x_{n_0},1)## . By hypothesis, the ball is a complete subspace and the subsequence ##\{x'_n\}## is cauchy. But then ##\{x'_n\}## converges to some point ##x \in K(x_{n_0},1)## This implies ##\{x'_n\}## converges to the same point x in the metric space X. So the original sequence ##\{x_n\}## has a convergent subsequence that converges to the point x, this means that ##\{x_n\}## converges to x. I've proved that an arbitrary cauchy sequence in X is convergent, this implies every cauchy sequence in X is convergent, then X is a complete metric space.


Is my proof correct?

Why are you having doubts? Eventually the tail of every cauchy sequence is contained in a closed ball. The closed ball is complete. So the sequence is convergent.
 
  • #3
222
0
Why are you having doubts? Eventually the tail of every cauchy sequence is contained in a closed ball. The closed ball is complete. So the sequence is convergent.

Thank you, I'm new to proofs and mathematics in general so I usually have strong doubts on whether I am doing the exercises right or not. I suppose it's a newbie's insecurity.
 
  • #4
Dick
Science Advisor
Homework Helper
26,260
619
Thank you, I'm new to proofs and mathematics in general so I usually have strong doubts on whether I am doing the exercises right or not. I suppose it's a newbie's insecurity.

Well, I thought that was a pretty fine and clear proof. I think you should start feeling less insecure.
 
  • #5
92
8
The only thing is that you don't want to say ε=1.

Because your metric space may be "smaller" than the unit closed ball.

You could just say for some ε. But the proof seems fine
 
  • #6
Dick
Science Advisor
Homework Helper
26,260
619
The only thing is that you don't want to say ε=1.

Because your metric space may be "smaller" than the unit closed ball.

You could just say for some ε. But the proof seems fine

A 'ball' in a metric space is just the set of all points x in the space such that |d(x,x0)|<=R. It doesn't have to be spherical. The proof really doesn't need any qualification.
 

Related Threads on Every closed ball in X is complete, then X is complete

Replies
6
Views
3K
Replies
5
Views
1K
Replies
8
Views
2K
Replies
2
Views
2K
Replies
2
Views
2K
  • Last Post
Replies
5
Views
9K
  • Last Post
Replies
1
Views
769
Replies
1
Views
774
  • Last Post
Replies
2
Views
1K
Top