> I've solved multiple continuous value problems by discretizing, applying combinatorics to the techniques, and then taking the limit of the result
But taking the limit of a sequence of rationals isn’t guaranteed to remain in the rationals (classic example: https://en.wikipedia.org/wiki/Basel_problem. Each partial sum is rational, but the limit of the partial sums is not)
So, how does that statement rebut “You can't do rigorous calculus (i.e. real analysis) on rationals alone.”?
> But taking the limit of a sequence of rationals isn’t guaranteed to remain in the rationals
I'm not saying it does. What I'm saying is that you can make a correspondence with the reals by using only rationals.
You can define convergence without invoking the reals (Cauchy convergence). If you take any such sequence, you give that sequence a name. That name is the equivalent of a real number. You can then define addition, multiplication - any operation on the reals - with respect to those sequences (again, invoking only rational numbers).
So far, we have two distinct entities: The rationals, and the converging sequences.
Then, if you want, you can show that if you take the rationals and those entities we're calling "converging sequences" together, you can make operations involving the two (e.g. adding a rational to that converging sequence) and eventually build up what we know to be the number line.