Equivalence of Definitions of Cyclic Group

$(1)$ implies $(2)$
Let $G$ be a Cyclic Group by definition 1.

Then by definition:
 * $(1): \quad g \in G$
 * $(2): \quad$ every element of $G$ is expressible as a power of $g$.

From $(1)$, it follows from Group Axion $G 1$: Closure that:
 * $\gen g \subseteq G$

From $(2)$ it follows that:
 * $G \subseteq \gen g$

Thus:
 * $G = \gen g$

and $G$ is a Cyclic Group by definition 2.

$(2)$ implies $(1)$
Let $G$ be a Cyclic Group by definition 2.

Then by definition:
 * $G = \gen g$

Thus as $g^1 \in \gen g$
 * $g \in G$

and by definition of generator:
 * $\forall h \in \gen g: h = g^n$

for some $n \in \Z$.

Thus $G$ is a Cyclic Group by definition 1.