[dawnofdusk]

In the diagonalization you don't need to assume the existence of any real numbers. Just on the left hand side you write down, formally, any sort of numbers that have decimal expansions that may be infinite. Rational numbers have infinite decimal expansions too, it's just that they will eventually repeat, but at this stage it's not necessary to think about what the properties of these infinite decimal expansions actually mean. Then the diagonalization argument shows that this set of numbers with infinite decimal expansions are uncountable and also contain the rationals. This still doesn't define the real numbers yet: to do so one needs to think about the Euclidean metric on the rationals and how to complete it.