Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

Let $(X,d)$ be an arbitrary complete metric space and suppose $S\subseteq(X,d)$. Show that $S$ is closed if and only if every Cauchy sequence in $S$ converges to a point in $S$.

I did the forward direction, is it correct?

Suppose $S$ is closed. Let $x\in S$ be an Cauchy sequence, $x=(x_n)_{n \in \mathbb N}$. Then, since $S$ is closed, it contains all of its limit points. Therefore, $\lim_{n\to\infty} (x_n)$ will converge to an element in $S$. So every Cauchy sequence in $S$ converges to an element in $S$.

For the backward direction, would I just let $S\subseteq(X,d)$ and suppose that every Cauchy sequence in $S$ converges to a point in $S$. Then show that a limit point $x$ is in $S$?

Any feedback is appreciated, thanks.

share|cite|improve this question
up vote 1 down vote accepted

The first direction is almost ok, but you have to argue why $(x_n)_n$ converges at all. Here you need, that $(x_n)_n$ is also a C-sequence in $(X,d)$ which is complete.

To show that S is closed, you have to show, that every in S convergent sequence has it limit value in S. But every convergent sequence is also a C-sequence...

share|cite|improve this answer
To finish the proof, by the assumption "every Cauchy sequence in S converges to a point in S", then the limit point of the convergent sequence is in $S$. Hence $S$ is closed. – Mhenni Benghorbal Nov 27 '12 at 7:36
@MhenniBenghorbal This comment was really not necessary. If Fant chose to leave this as an indication, leave it as such. – Did Dec 1 '12 at 9:35

Prove the contrapositive for the other direction. Assume $S$ is not closed and find a Cauchy sequence in $S$ which does not converge in $S$ (not closed means that one limit point is not there in the set, so construct a Cauchy sequence which converge to this limit point).

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.