Equivalence of Definitions of Kuratowski Closure Operator

Definition 2 implies Definition 1
A closure operator, by definition, is inflationary and idempotent.

Thus it follows immediately that Definition 2 implies Definition 1.

Definition 1 implies Definition 2
Let $X$ be a set.

Let $\operatorname{cl}$ be a Kuratowski closure operator on $X$ by Definition 1.

By definition of closure operator, it remains to be proved that $\operatorname{cl}$ is increasing.

Let $A \subseteq B \subseteq X$.

Then by Definition 1 and Union with Superset is Superset:
 * $\operatorname{cl} \left({B}\right) = \operatorname{cl} \left({A \cup B}\right) = \operatorname{cl} \left({A}\right) \cup \operatorname{cl} \left({B}\right)$

By Set is Subset of Union:
 * $\operatorname{cl} \left({A}\right) \subseteq \operatorname{cl} \left({A}\right) \cup \operatorname{cl} \left({B}\right) = \operatorname{cl} \left({B}\right)$