How did Georg Cantor's diagonal argument show that real numbers cannot be listed?

Georg Cantor used his diagonal argument to demonstrate that the set of real numbers is uncountable, meaning they cannot be put into a one-to-one correspondence with the integers.12

The proof works by contradiction. Suppose you could list all real numbers between 0 and 1 in an infinite sequence. Cantor showed that you could always construct a new real number that is not on your list. By looking at the digits of the numbers in your list—specifically the $n$-th digit of the $n$-th number—you can create a new number by changing each of those digits (for example, by adding 1 to each digit and taking the result modulo 10).

Because this new number differs from the first number in the first decimal place, from the second number in the second decimal place, and so on for every number in the sequence, it cannot be any of the numbers already listed. Thus, the real numbers are "uncountable" or nondenumerable, proving that there are different sizes of infinity.123