If your real numbers are Cauchy sequences of rational numbers, and your rational numbers are a subset of real numbers, then your rational numbers are a subset o

If your real numbers are Cauchy sequences of rational numbers, and your rational numbers are a subset of real numbers, then your rational numbers are a subset of Cauchy sequences of rational numbers (which violates the axiom of foundation). This is not as trivial as it sounds, which is why we invented all these different tools for explaining what “is” or “equals” means in mathematics without resorting to set equality (equality, isomorphism, equivalence, etc).