Suppose is an accumulation point of a set . Then there existsdistinctsuch that . Let then by definition of convergence it means for . Thus, aredistinctelements which lie in and hence there are infinitely many.

A set is closed if and only if it contains all its accumulation points. A finite set clearly has no accumulation points, hence the set is empty. And it contains the empty set.Using this, prove that any finite subset of X is closed.