Coming across diagonalization, I was thinking of other methods to disprove the existence of a bijection between reals and naturals. Can any method that shows that a completely new number is created which is different from any real number in the list disprove the existence of a bijection?
For example, assume we have a full list in some order of real numbers. Take two adjacent numbers and calculate their average, which adds a digit to the end of the number. That number is not on the list. Does this suffice?