Question:
Show that $A$ is a subsemigroup of $S$ if and only if $A^{2}\subset A$. The subset here may not necessarily be proper.
My approach,
Suppose $A$ is a subsemigroup of $S$, then for all $x,x\in A, x^{2}\in A$. Does this mean that, $A^{2}\subset A$ since $(x,x)\in A^{2}$?. I am not sure!
Conversely, suppose $A^{2}\subset A$, then for any $(x,x)\in A^{2}$, $x^2\in A$ since $A^{2}\subset A$. This implies $A$ is a subsemigroup of $S$.
I am trying to convince myself with this proof, but it seems to me that what I did is not a good approach. Can somebody help me?