Cauchy Sequences in a discrete space

Question

How can I find all Cauchy sequences in a discrete space? Any hint would be helpful since I cannot vision this in my mind. Thank you in advance.

Edit: Thanks for the responses. As far I understand (though not sure) discrete space (X,d) is a metric space where all its subsets are open in X. I think discrete metric generates discrete spaces, but so does other metrics. So I am still trying to find an answer:( Correct me if I am wrong though:(

Edit2: Hi, I wanted to give an update for future generations. It seems that this question is asked in the wrong way. While this question is valid if it is asked in "a space with discrete metric" if it is asked with "a discrete metric space" the answer gets nasty. So I was given a wrong wording thats why I was confused.

Thanks a lot for the help. Hope no one would have to deal with the second one :D

Answer 1

HINT: By discrete space I assume that you mean metric space with the discrete metric. Let $X$ be such a space, and let $\langle x_n:n\in\Bbb N\rangle$ be a Cauchy sequence in $X$. Then there is an $m\in\Bbb N$ such that $d(x_k,x_\ell)<1$ whenever $k,\ell\ge m$. (Why?) If $d$ is the discrete metric, and $d(x_k,x_\ell)<1$, what can you conclude about $x_k$ and $x_\ell$?

Answer 2

These are sequences such that there exists $N$ such that $x_{n+N}=x_N$.