Non-Trivial Group has Non-Trivial Cyclic Subgroup

Theorem
Every group has at least one cyclic subgroup.

Proof
If $$g$$ has infinite order, then $$\left \langle {g} \right \rangle$$ is an infinite cyclic group.

If $$\left|{g}\right| = n$$, then $$\left \langle {g} \right \rangle$$ is a cyclic group with $$n$$ elements.

This follows from the definition of the order of an element and the definition of cyclic group.