# 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.

$\blacksquare$
The text passage is a formal mathematical proof that the set of rational numbers is countably infinite. It does not contain any explicit mathematical questions or answers to such questions.
no_question
Given a text passage, determine:
    1. If it contains a mathematical question (including word problems or calculations)
    2. If an answer to that mathematical question is provided in the text

First, describe the content structure you found in 1-2 sentences.

Then provide your analysis in the following format:
TYPE: <one of the following values>
- NO_QUESTION
- SINGLE_QUESTION
- SINGLE_QUESTION_WITH_ANSWER
- MULTIPLE_QUESTIONS
- MULTIPLE_QUESTIONS_WITH_ANSWERS
system
# 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.

$\blacksquare$
user
The text passage is a formal mathematical proof that the set of rational numbers is countably infinite. It does not contain any explicit mathematical questions or answers to such questions.

TYPE: NO_QUESTION