# show that in a metric space a Cauchy sequence converge

show that in a metric space a Cauchy sequence converges if and only if it has a convergent subsequence.

Definition: A sequence ${P_ {n}}$ in a metric space is called Cauchy sequence if $(\forall \epsilon> 0) (\exists N\in \aleph) (\forall m, n \geq N)$, $d (p_ {n}, p_ {m}) <\epsilon$

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You have to do more than copy a question out of a textbook. –  Nate Eldredge Apr 25 '12 at 4:19
@ChalieHer Every Cauchy is convergent? (Hint: Is every metric space complete?) –  user21436 Apr 25 '12 at 4:29
No, you don't know that every Cauchy sequence is convergent. (or, if you think you know that, then you "know" something false; to quote Mark Twain, "It's not what we don't know that hurts us, it's what we 'know' that just ain't so".) That's true in the reals, but it's not true in an arbitrary metric space. The definition of Cauchy sequence says nothing about convergence. What is the definition? –  Arturo Magidin Apr 25 '12 at 4:32
You folks are very patient. –  copper.hat Apr 25 '12 at 4:42

Number one you have dropped your homework problem on us as if you're expecting everyone else to solve it for you. Please don't do this. Now your problem is very easy once you know the definitions and here are some hints that should be enough for you to work out the whole problem.

One direction is a triviality (which direction?). For the other direction I suggest you consider the following: If you have a convergent subsequence $p_{n_k}$ that converges to $p$, consider the quantity

$$d(p_n+ p_{n_k}, p + p_{n_k}).$$

Do you know the triangle inequality? This trivialises the problem if you already know the definition of a cauchy sequence.

Let me give you the definition of a cauchy sequence: Let $(X,d)$ be a metric space. We say that a sequence $x_n$ in $X$ is cauchy if for all $\epsilon>0$, there exists a natural number $N$ such that $m,n \geq N$ implies that $d(x_n,x_m) < \epsilon.$

Supplementary question: A metric space $X$ is said to be sequentially compact if every sequence $x_n$ that lives in $X$ has a convergent subsequence. Prove that any cauchy sequence in a sequentially compact metric space is convergent.

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