In high-school calculus (if you've taken that), you apply things like "d/dx" which LOOKS like it is just a fraction, dividing "d" by "d times x". The notation was first dreamed up by mathematicians who were thinking "but if we keep making the d-slices really REALLY small we would move from the discrete approximation to the formula for the correct continuous answer".

Unfortunately, while SOME such formulas worked (the ones you learned in calculus - stuff like the chain rule), other formulas generated using the same kind of reasoning ("just imagine that d becomes infinitely small") come up with answers that are nonsensical or just wrong. How can we know when it's OK to just say "let things get infinitely small" and when it gives bogus answers?

The solution, which is taught in high-school calculus, was the "epsilon-delta" formulation. Instead of saying "let d become infinitely small" (a statement that may just be nonsense), say "I will prove that for ANY small epsilon (greater than 0) I can find a delta > 0 such that for values of x closer than delta, the error will be less than epsilon". That statement doesn't require an infinity to exist anywhere -- it is just a statement about particular finite numbers. And we can build calculus on such principles.

This isn't an EXACT analogy to your question about real numbers, but it uses the same type of reasoning, and I'm hoping the analogy is in terms of mathematical reasoning you are already familiar with.