Artinian implies periodic

Statement
An Artinian group (i.e., a group in which every descending chain of subgroups stabilizes at a finite stage) must be a periodic group: every element in the group has finite order.

Proof
We prove the contrapositive.

Given: A group $$G$$ and an element $$g \in G$$ of infinite order.

To prove: $$G$$ is not Artinian.

Proof: Consider the descending chain of subgroups:

$$\langle g \rangle \ge \langle g^2 \rangle \ge \langle g^4 \rangle \ge \dots $$.

Since $$g$$ has infinite order, this is a strictly descending chain of subgroups that never stabilizes. Thus, $$G$$ is not Artinian.