Finite Union of Finite Sets is Finite/Proof 1

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 $S = \O$, so $\bigcup S = \O$, which is finite.

Suppose that an arbitrary finite set with cardinality $n$ of finite sets has a finite union.

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

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

Then:

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

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

$\blacksquare$

Sources

• 2000: James R. Munkres: Topology (2nd ed.) ... (previous) ... (next): $1$: Set Theory and Logic: $\S 6$: Finite Sets: Corollary $6.8$