If truly interested, one can find an undergraduate textbook for "real analysis" and read the first portion of it (no need to read the whole book), and see the modern treatment or resolution of these intuitive or "ancient" concepts.
Without the acquirement of the more abstract modern notions, this topic can appear more confusing and more difficult than it should necessarily be.
Given stonebench's intelligence, he could acquire it effectively in two weeks, or maybe just one.