See cyclic group, order of a group

Let $G$ be a group, and let $a\in G$.
If $|a|=\mathrm{\infty }$, then $|⟨a⟩|=\mathrm{\infty }$

Proof

Recall that $⟨a⟩=\{a^k:k\in \mathbb{Z}\}$
Every power of a group element with infinite order is unique.
In other words, since $a^i=a^j$ if and only if $i=j$, each $a^k$ is distinct.
Since $\mathbb{Z}$ is infinite, $⟨a⟩$ is infinite.

QED