To fill in some details: Two real numbers are different if in the same place they have different digits. [1] So the idea is to take your infinite, hypothetical

To fill in some details: Two real numbers are different if in the same place they have different digits. [1] So the idea is to take your infinite, hypothetical list of all real numbers A[] and make a new real number X, where for all i, there's some digit where A[i] and X differ. Easiest way is to make X differ from A[i] at digit i. [1] There's a subtlety here about repeating nines at the end of a number, but it is inessential.