Cancellable Infinite Semigroup is not necessarily Group

Theorem
Let $\left({S, \circ}\right)$ be a semigroup whose underlying set is infinite.

Let $\left({S, \circ}\right)$ be such that all elements of $S$ are cancellable.

Then it is not necessarily the case that $\left({S, \circ}\right)$ is a group.

Proof
Consider the semigroup $\left({\N, +}\right)$.

From Natural Numbers under Addition form Commutative Semigroup, $\left({\N, +}\right)$ forms a semigroup.

From Natural Numbers are Infinite, the underlying set of $\left({\N, +}\right)$ is infinite.

From Natural Number Addition is Cancellable, all elements of $\left({\N, +}\right)$ are cancellable.

But from Natural Numbers under Addition do not form Group, $\left({\N, +}\right)$ is not a group.