Let X be a set donated by the discrete metrics
d(x,y) = 0 if x=y,
1 if x̸=y.
Show that a subset Y of X is compact iff this set is closed
I don't understand. Maybe I am speaking out of turn here, but isn't every subset of $\displaystyle X$ closed? For, to claim that it wasn't closed would be to say that the subset doesn't contain one of it's limit points but no point in $\displaystyle X$ has a limit point (as can be seen by taking a neighborhood of radius one-half). Did you mean finite?
First, notice $\displaystyle \left(x_n\right)_{n\in\mathbb{N}}\subset X$ converges iff it becomes constant after a certain $\displaystyle N\in\mathbb{N}$. With that, every subset is closed.
Now, im not sure about this excercise. I think $\displaystyle Y$ must be finite... if not we may choose $\displaystyle X=\mathbb{N}$, $\displaystyle Y=X-\{1\}$ and the sequence $\displaystyle x_n=n+1.$ that doesn't have any convergent sub-sequence 'cause it never become constant.