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 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 has a limit point (as can be seen by taking a neighborhood of radius one-half). Did you mean finite?
First, notice converges iff it becomes constant after a certain . With that, every subset is closed.
Now, im not sure about this excercise. I think must be finite... if not we may choose , and the sequence that doesn't have any convergent sub-sequence 'cause it never become constant.