Maths query

[OwlHoot]

The reason they are the same number is nothing to do with limits; rather the definition of real numbers.

In classical maths, which covers most requirements, the real numbers can be rigorously defined in terms of equivalence classes of Cauchy sequences, and that involves limits.

The main reason I emphasised limits, distinguishing them from sequences, was that confusion over the two is the main reason these "does 0.999.. equal 1?" debates crop up over and over again on unmoderated maths & tech forums.

Your "axiomatic" approach might satisfy a professional; but it doesn't help dispel obstinately held misconceptions, about what is really no more than a matter of notation once the relevant concepts are clearly grasped.