Hausdorff's forgotten proof that almost all numbers are normal

TL;DR

This paper revisits Hausdorff's 1914 proof and extends it to demonstrate that almost all real numbers are normal, providing an elementary and accessible proof suitable for undergraduates.

Contribution

The paper generalizes Hausdorff's original proof to establish that almost all numbers are normal, offering a simpler and more accessible proof than previous methods.

Findings

Almost all numbers are normal in base 2.

The proof is elementary and suitable for undergraduate students.

Provides a new perspective on classical normality results.

Abstract

In 1914, Felix Hausdorff published an elegant proof that almost all numbers are simply normal in base 2. We generalize this proof to show that almost all numbers are normal. The result is arguably the most elementary proof for this theorem so far and should be accessible to undergraduates in their first year.