This structured list is the basis for constructing a specific real number that is logically guaranteed to be missing from the sequence, thereby invalidating the initial premise of countability.

The contradiction is formed when defining this new real number, , where each -th decimal digit is chosen specifically to differ from the -th digit of the -th number in the list. By ensuring each digit is not equal to (while avoiding 0 or 9 to prevent ambiguities with repeating decimals), the resulting number is distinct from every element in