Element has Idempotent Power in Finite Semigroup

Theorem
Let $\struct {S, \circ}$ be a finite semigroup.

For every element in $\struct {S, \circ}$, there is a power of that element which is idempotent.

That is:
 * $\forall x \in S: \exists i \in \N: x^i = x^i \circ x^i$

Proof
From Finite Semigroup Equal Elements for Different Powers, we have:


 * $\forall x \in S: \exists m, n \in \N: m \ne n: x^m = x^n$

Let $m > n$.

Let $n = k, m = k + l$.

Then:
 * $\forall x \in S: \exists k, l \in \N: x^k = x^{k + l}$

Now we show that:
 * $x^k = x^{k + l} \implies x^{k l} = x^{k l} \circ x^{k l}$

That is, that $x^{k l}$ is idempotent.

First:

From here we can easily prove by induction that:
 * $\forall n \in \N: x^k = x^{k + n l}$

In particular:
 * $x^k = x^{k + k l} = x^{k \paren {l + 1} }$

There are two cases to consider:


 * $(1): \quad$ If $l = 1$, then $x^k = x^{k \paren {l + 1} } = x^{2 k} = x^k \circ x^k$, and $x^{k l} = x^k$ is idempotent.


 * $(2): \quad$ If $l > 1$, then:

and again, $x^{k l}$ is idempotent.