Semigroup is Subsemigroup of Itself

Theorem
Let $\left({S, \circ}\right)$ be a semigroup.

Then $\left({S, \circ}\right)$ is a subsemigroup of itself.

Proof
For all sets $S$, $S \subseteq S$, that is, $S$ is a subset of itself.

Thus $\left({S, \circ}\right)$ is a semigroup which is a subset of $\left({S, \circ}\right)$, and therefore a subsemigroup of $\left({S, \circ}\right)$.