Real Analysis Connections Map

Real Analysis · Axiom Academy

SUMMARY Real Analysis Connections Map One property — the completeness of — quietly underwrites every major theorem of the course. Here is the whole dependency map. Real analysis is one unified theory, not a pile of separate theorems — and its single foundation is the completeness (least-upper-bound) property of . Completeness feeds sequences (every Cauchy sequence converges; bounded-monotone sequences hit their sup), which in turn power series and the sequential view of continuity . Add the right topology and the big theorems appear: continuity on a compact set gives the EVT; continuity on a connected interval gives the IVT. The chain finishes in calculus: EVT → Rolle → the MVT , and continuity + the MVT underwrite the FTC , which links differentiation and integration. Topology — open / closed / compact / connected / metric — is the unifying language that lets these ideas travel beyond . Each arrow reads is built on / underwrites : it points from a result to the result that depends on it. Follow any path back and it ends at completeness. 11 ideas, 14 dependencies — every one traceable to the least-upper-bound axiom. The Foundation Completeness of Every nonempty set of reals that is bounded above has a least upper bound . This one axiom is what separates from (which has holes ), and it is the root of the entire map. Why it matters: it manufactures existence — limits, suprema, and roots that cannot guarantee.

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