Finite Union of Finite Sets is Finite/Proof 2

Theorem

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

Then the union of $S$ is finite.

Proof

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

Set:

$m = \max \set {\card {A_1}, \ldots, \card {A_n} }$

Then:

$\ds \card {\bigcup_{k \mathop = 1}^n A_k} \le \sum_{k \mathop = 1}^n \card {A_k} \le \sum_{k \mathop = 1}^n m = n m$

This article, or a section of it, needs explaining. In particular: needs to invoke a link to result that shows $\card {\bigcup_{k \mathop = 1}^n A_k} \le \sum_{k \mathop = 1}^n \card {A_k}$ You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by explaining it. To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{Explain}} from the code.

Hence the result.

$\blacksquare$

Also see

• Inclusion-Exclusion Principle