Uncountability of Numbers with Restricted Decimal Expansion

Prove that the set of real numbers between 0 and 1, where their decimal expansion only contains the digits 3 and 4, is uncountable using Cantor's diagonalization argument.

In this problem, we delve into the fascinating realm of uncountable sets, a key concept in real analysis and set theory. The task is to show that the set of numbers between zero and one, whose digits in their decimal expansions are restricted to threes and fours, is uncountable. This is a direct application of Cantor's diagonalization argument, a powerful technique for proving the uncountability of certain sets.

Cantor's diagonal argument is foundational because it illustrates a systematic way to construct an element not in a given list, demonstrating that no list could possibly enumerate all elements of the set. In this case, suppose we have a countable list of all numbers whose decimals are composed only of threes and fours. By altering the nth digit of the nth number in this list, creating a new number not in the original list, we reveal a contradiction. This crafty logic underscores the richness and complexity of real analysis and shines light on the infinite nature of real numbers, even within tight confines like restricted decimal expansions.

The exercise deepens understanding of countable versus uncountable sets, shedding light on the characteristics that lead to uncountability. Knowing how to apply Cantor's diagonalization helps build a robust framework for dealing with more complex problems in set theory and other areas of mathematical analysis, making this a valuable problem for students to engage with.