Infinite Set of Natural Numbers is Countably Infinite

Theorem

Let $\N$ be the set of natural numbers.

Let $S$ be an infinite subset of $\N$.


Then $S$ is countably infinite.

That is, there is a bijection $f: \N \to S$.


Proof

By Infinite Set has Countably Infinite Subset, we have an injection $g: \N \to S$

But by Cantor-Bernstein-Schröder Theorem/Lemma this produces a bijection $f: \N \to S$

$\blacksquare$

Sources

  • 2000: James R. Munkres: Topology (2nd ed.) ... (previous) ... (next): $1$: Set Theory and Logic: $\S 7$: Countable and Uncountable Sets: Theorem $7.1$
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