Countably Infinite Set has Enumeration

This article needs proofreading. Please check it for mathematical errors. If you believe there are none, please remove {{Proofread}} from the code. 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 {{Proofread}} from the code.

Theorem

Let $S$ be a countably inifnite set.

Then there exists a countably infinite enumeration $\set{s_1, s_2, s_3, \ldots}$ of $S$.

Proof

By definition of countably inifnite set:

there exists a bijection $f:S \to \N$

From Inverse of Bijection is Bijection:

$f^{-1} : \N \to S$ is a bijection

Let $s = f^{-1}$.

It follows that $s : \N \to S$ is a countably infinite enumeration $\set{s_1, s_2, s_3, \ldots}$ by definition.

$\blacksquare$