Union of g-Tower is Greatest Element and Unique Fixed Point

Theorem
Let $M$ be a set.

Let $g: M \to M$ be a progressing mapping on $M$.

Let $M$ be a $g$-tower.

Then:
 * $\ds \bigcup M = M$
 * $M$ is the greatest element (under the subset relation) of $M$
 * $M$ is a fixed point of $M$.