Rational Numbers are Countably Infinite/Proof 3
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Theorem
The set $\Q$ of rational numbers is countably infinite.
Proof
For each $n \in \N$, define $S_n$ to be the set:
- $S_n := \set {\dfrac m n: m \in \Z}$
By Integers are Countably Infinite, each $S_n$ is countably infinite.
Because each rational number can be written down with a positive denominator, it follows that:
- $\forall q \in \Q: \exists n \in \N: q \in S_n$
which is to say:
- $\ds \bigcup_{n \mathop \in \N} S_n = \Q$
By Countable Union of Countable Sets is Countable, it follows that $\Q$ is countable.
Since $\Q$ is manifestly infinite, it is countably infinite.
$\blacksquare$
Sources
- 2008: Paul Halmos and Steven Givant: Introduction to Boolean Algebras ... (previous) ... (next): Appendix $\text{A}$: Set Theory: Countable Sets