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.
  
  $\blacksquare$
explanation: The document is a mathematical proof that demonstrates the theorem stating that the set of rational numbers ($\Q$) is countably infinite. It does not contain a question or an answer to a question.
distilabel_metadata.content: Evaluate the user given text extract and determine if it contains a math question and optionally and answer to this question.
  
  Output rules:
  
  1. Briefly describe the document and mention if it contains a question or a question and answer.
  2. Finalize the output with: Label: no-question,question,question-and-answer,multiple-question,multiple-question-and-answer
distilabel_metadata.role: system
distilabel_metadata.content: # 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.
  
  $\blacksquare$
distilabel_metadata.role: user
distilabel_metadata.raw_output_q_a_detector_0: The document is a mathematical proof that demonstrates the theorem stating that the set of rational numbers ($\Q$) is countably infinite. It does not contain a question or an answer to a question.
  
  Label: no-question