infinite subset of discrete metric space is not compact

Question

The question is

Im not really sure how to go about this

So far i am trying to show that for an open cover of the infinite subset X, there isn't a finite sub cover and therefore X is not compact

I am not really sure how to show this properly.

Answer 1

You shouldn't be "trying to show that for an open cover of the infinite subset $X$, there isn't a finite subcover"; rather you should be trying to show that for each infinite set $X$ of points there is at least one open cover of $X$ that has no finite subcover. The negation of the statement $\text{“}$For every open cover blahblahblah$\text{''}$ is the statement $\text{“}$For at least one open cover, not blahblahblah.$\text{''}$

For each point in $X$, you can find an open neighborhood of that point that contains no other point of the space; hence no other point of $X$. That's where you use the fact that the space is discrete.

Can you take it from there?