Proof:The set of reals is uncountable
We can put the real numbers into a common format by listing out their digits:
Now, cannot be in the image of , because for any , differs from in the th digit. This contradicts the fact that is surjective, completing the proof.
We construct a new diabolical number by changing each digit on the diagonal. We can add 5 to each digit (wrapping around if necessary) to set the corresponding digit of . In example table above, we would have .