text: # Rational Numbers are Countably Infinite/Proof 3
  
  ## 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.
  
  $\blacksquare$