The problem with mapping the reals to the integers is that a list of real numbers is infinite in "two directions". Each real can be potentially described with an infinite number of digits, and you can have an infinite list of real numbers (i.e. infinite sequences of digits). This is, I believe, the basis for Cantor's diagonal argument, which is a quite beautiful proof of there being more reals than integers.

https://en.wikipedia.org/wiki/Cantor%27s_diagonal_argument