Finite Union of Finite Sets is Finite/Proof 2

Proof
Let $S = \left\{{A_1, \ldots, A_n}\right\}$ such that $A_k$ is finite $\forall k = 1, \ldots, n$.

Set:
 * $m = \max \left\{{ \left|{A_1}\right|, \ldots, \left|{A_n}\right|}\right\}$

Then:
 * $\displaystyle \left|{ \bigcup_{k \mathop = 1}^n A_k}\right| \le \sum_{k \mathop = 1}^n \left|{A_k}\right| \le \sum_{k \mathop = 1}^n m = n m$

Hence the result.

Also see

 * Inclusion-Exclusion Principle