Understanding Rudin's Proof of the Density of Rational Numbers

Identifying the importance of the density of the rational numbers within the real numbers is fundamental in real analysis. Rudin's proof of this fact is both elegant and straightforward, relying on the properties of real numbers and rational numbers. This article will delve into the proof and provide a clear understanding of its components, ensuring that readers gain a comprehensive grasp of this important theorem.

Statement of the Theorem

The theorem states that for any two real numbers (a

Outline of the Proof

Understanding the Setup

We begin by considering two arbitrary real numbers (a) and (b) such that (a

Using the Density of Integers

Since (b > a), the difference (b - a) is a positive real number. We choose a positive integer (n) such that (frac{1}{n}

Bounding Integers

Given (n), we consider the numbers (na) and (nb). Since (a [n a Moreover, since (frac{1}{n} [n (b - a) n b - n a > 1]

This implies that there are at least two integers (k_1) and (k_2) such that:

[k_1 leq n a These (k_1) and (k_2) are consecutive integers.

Finding the Rational

The integer (k_1) satisfies:

[k_1 leq n a Dividing through by (n) yields:

[frac{k_1}{n} leq a This means:

[a Thus, we can set:

[r frac{k_1 1}{n}]

This (r) is a rational number because both (k_1) and (n) are integers.

Conclusion

Therefore, for any (a

Summary

Rudin's proof relies on the properties of real numbers and rational numbers, particularly the ability to find integers that can be bounded within any interval. By manipulating these integers and their ratios, we can always find a rational number within any two real numbers, thus demonstrating the density of the rationals in the reals.

Additionally, it is worth noting that any real number (x) can be associated with an open interval (O) that necessarily contains a rational number because the rationals are arbitrarily close to all real numbers. This is evident from the decimal expansion of (x).