Can $ \ \ \sup_{x \in X} \inf_{y \in Y} \ d(x,y) \neq 0$?

Question

Let $X, \ Y$ be two non-empty subset of a metric space $ (M,d)$ such that $X \subset Y$.

My question is-

Can $ \ \ \sup_{x \in X} \inf_{y \in Y} \ d(x,y) \neq 0$ ?

My calculation shows that always $ \ \ \sup_{x \in X} \inf_{y \in Y} \ d(x,y) = 0$

Answer 1

$inf_{y\in Y}d(x,y)=0$ since $x\in Y$ therefore the sup is always zero.