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

Proof
Let the hypothesis be assumed.

By definition, $M$ is a nest which is also a set

Hence by definition $M$ is a chain.

By $g$-Tower is Closed under Chain Unions:
 * $\ds \bigcup M \in M$

It follows directly that $\ds \bigcup M$ is the greatest element (under the subset relation) of $M$

Hence $\ds \bigcup M$ cannot be a proper subset of $\ds \map g {\bigcup M}$.

Thus:
 * $\ds \bigcup M = \map g {\bigcup M}$

and the result follows.