Countability of Rational Numbers Using Classical Method

Prove that the set of rational numbers $\mathbb{Q}$ is countable by using the classical method of defining the height of a rational number and showing it as a countable union of finite sets.

In this problem, we explore the countability of the set of rational numbers, a fundamental concept in real analysis that distinguishes between different "sizes" of infinity. The key strategy employed is the classical method of defining the height of a rational number. The height of a rational number is usually defined as the sum of its absolute numerator and denominator, assuming it is expressed in the simplest form. This technique allows us to categorize rational numbers into different groups based on this height parameter. The rational numbers can then be perceived as a countable union of these finite sets, each corresponding to a different height.

This method not only provides a proof for the countability of rational numbers, but it also helps illuminate the power of using a systematic ordering or structuring principle in proofs related to infinite sets. By recognizing that even though there are infinitely many rational numbers, they can be ordered in a sequence that pairs each with a natural number, we establish their countability.

Understanding this proof is crucial because it exemplifies a standard approach to demonstrating the countability of sets, which can be applied in various contexts within and beyond real analysis. The concept of countability is particularly foundational when dealing with different types of infinity and is essential for building intuition about the structure and nature of real numbers, which plays a significant role in the broader study of analysis.