Prove that any cyclic monoid is either isomorphic to (N, +) or is isomorphic to a monoid of the form of a finite cyclic monoid of some size.
I understand that this is saying that a cyclic monoid is either infinite or finite but I don't know how to rigorously prove this. Any help would be great, thank you in advance!
Hint: Let $(M,\circ,e)$ be a cyclic monoid with unit element $e$. Then there is an element $m\in M$ such that $M=\{m^k\mid k\in{\Bbb N}_0\}$. There are two cases. Either $m^k\ne e$ for each $k\geq 1$ or $m^k=e$ for some minimal $k\geq 0$.