Infinite Pigeonhole

Last updated Nov 1, 2022

Let $X$ be an Infinite Set. Let $\bigsqcup\limits_{i \in [n]} A_{i} = X$ be a finite Partition of $X$ for $n \in \mathbb{N}$. Then $\exists j \in [n]$ s.t. $A_{j}$ is infinite.

Suppose not. Then each $A_{i}$ is finite for $i \in [n]$. Denote $|A_{i}|= c_{i}$. Since $A_{i}$ are Mutually Disjoint, we have that $\infty > \sum\limits_{i \in [n]} c_{i} = \big|\bigsqcup\limits_{i \in [n]} A_{i} \big| = |X|$, contradicting our assumption that $X$ was infinite. $\blacksquare$

• Cardinal Arithmetic