Finite Union of Finite Sets is Finite/Proof 1

Proof
The proof proceeds by induction.

Let $S$ be a finite set with cardinality $n$.

If $n = 0$ then $S = \O$, so $\bigcup S = \O$, which is finite.

Suppose that the union of any finite set of finite sets with cardinality $n$ has a finite union.

Let $S$ have cardinality $n^+$.

Then there is a bijection $f: n^+ \to S$.

Then:
 * $\ds \bigcup S = \bigcup_{k \mathop \in n^+} \map f k = \bigcup_{k \mathop \in n} \map f k \cup \map f n$

By Union of Finite Sets is Finite, $\bigcup S$ is finite.