Element of Finite Group is of Finite Order

Theorem
In any finite group, each element has finite order.

Proof
Let $G$ be a group whose identity is $e$.

From Finite Semigroup Exists Idempotent Power, for every element in a finite semigroup, there is a power of that element which is idempotent.

As $G$, being a group, is also a semigroup, the same applies to $G$.

That is:
 * $\forall x \in G: \exists n \in \N^*: x^n \circ x^n = x^n$.

From Identity Only Idempotent Element in Group, it follows that:
 * $x^n \circ x^n = x^n \implies x^n = e$.

So $x$ has finite order.

Alternative Proof
Follows as a direct corollary to the result Powers of Infinite Order Element.