How does Cantor’s diagonal argument show that real numbers cannot be completely listed?

Georg Cantor’s diagonal argument is a foundational proof in set theory used to demonstrate that the set of real numbers is uncountable, meaning its cardinality is strictly greater than that of the integers or counting numbers.123

The argument functions by contradiction:
* Assume that all real numbers between 0 and 1 could be listed in an infinite sequence, pairing each one with a unique counting number (1, 2, 3, ...).23
* Cantor showed that even in such a list, one could construct a new real number that is not included.2
* By looking at the digits along the "diagonal" of this list—where the first digit of the first number is changed, the second digit of the second number is changed, and so on—a new number is created that differs from every number in the original list by at least one decimal place.2

Because this new number cannot be part of the list, the initial assumption that the real numbers can be completely listed is false. This proves that there are different "sizes" of infinity, establishing that the continuum of real numbers is a larger transfinite cardinal than the set of counting numbers.2314