Countably Infinite Set has Enumeration
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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.
$\blacksquare$