Finite Union of Finite Sets is Finite

Theorem
Let $S$ be a finite set of finite sets.

Then the union of $S$ is finite.

Proof
The proof proceeds by induction.

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

If $n = 0$ then $S = \varnothing$, so $\bigcup S = \varnothing$, 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:
 * $\bigcup S = \bigcup_{k \in n^+} f \left({k}\right) = \bigcup_{k \in n} f \left({k}\right) \cup f \left({n}\right)$

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