# Infinite set which is Dedekind finite and Weierstrass compactness

Weierstrass compactness states that each infinite set has a limit point. Why Infinite set which is Dedekind finite with discrete metric not Weierstrass compact.

Recall that $x$ is a limit point of $A$ if and only if every neighborhood of $x$ meets $A$ on an infinite set. So in any case a discrete space will never be Weierstrass compact.
Simply because $\{x\}$ is a neighborhood of $x$. So no infinite set has any limit point. So the only way a discrete space can be Weierstrass is that it is finite, and the requirement is satisfied vacuously.