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: $$\forall x \in G: \exists n \in \mathbb{N}^*: x^n \circ x^n = x^n$$.

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

So $$x$$ has finite order.