Infinite Set of Natural Numbers is Countably Infinite
Jump to navigation
Jump to search
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 $7.1$