Problem:

Let {

} be a Cauchy sequence in a metric space (M,D). Let A={

, ...}.

Suppose that {

} doesn't converge in M. Prove that A is a closed subset of (M,D).

What I have done:

So we know {

} is Cauchy, so let

>0 be given. Then there exists a positive integer N such that the D(

)<

for all i,j

N. Also, and this may be where I'm stuck.. Since {

} doesn't converge in M, then we know that for any y

M,

>0, and N, we can find an n>N such that D(

, y)>

.

That last statement could be wrong/unnecessary, so any help you can give me on where to go would be very helpful.