Let $n\ge 2$ be a natural number, and $A$ be a semigroup with more than $2$ elements. Let every subset of $A$ with n elements is a subsemigroup of A. Prove that every subset of $A$ is a subsemigroup of $A$.
If $n=2$ the answer is clear. But what if $n>2$ since we don't know either $A$ finite or not?