I don't think this construction works, for the same reason that rationals are lower cardinality than reals.
I e, I think your program proves the rationals, not the reals.
They would be rationals if the "lists" were finite (i.e. if their input were finite, like "natural numbers less than 100"). Infinite lists of digits (i.e. where the input can be any natural number) are reals.
One subtlety is that we're talking about arbitrary functions, e.g. they aren't necessarily computable, or even representable (e.g. as programs, or otherwise).
One subtlety is that we're talking about arbitrary functions, e.g. they aren't necessarily computable, or even representable (e.g. as programs, or otherwise).