Real Analysis | Supremum, Completeness Axiom & Why Real Numbers Are Special

Описание: What makes the real numbers COMPLETE? Why do the rationals have "holes"? And how does one axiom make all of calculus possible?

In this video, we dive deep into Chapter 1 of Tom Apostol's Mathematical Analysis, covering:

📐 SECTION 1.10 — Upper Bounds, Maximum Element & Supremum
What is an upper bound vs. THE least upper bound?
Why the supremum might not be in the set
The crucial difference between maximum and supremum

⚡ SECTION 1.11 — The Completeness Axiom (Axiom 10)
The most important axiom in all of analysis
Why ℚ has gaps but ℝ doesn't
How √2 reveals the incompleteness of rationals

🔧 SECTION 1.12 — Properties of the Supremum
Theorem 1.14: The Approximation Property
Theorem 1.15: The Additive Property (sup of sums)
Theorem 1.16: The Comparison Property

🔢 SECTION 1.13 — Integer Properties from Completeness
Theorem 1.17: The integers are unbounded
A beautiful proof using the completeness axiom

This is Part 3 of my Real Analysis series based on Apostol's textbook — designed for math majors and anyone transitioning to proof-based mathematics.

═══════════════════════════════════════

📚 TEXTBOOK: Mathematical Analysis (2nd Edition) by Tom M. Apostol

═══════════════════════════════════════

🎯 WHO IS THIS FOR?
Math majors taking Real Analysis
Graduate students preparing for qualifying exams
Self-learners working through Apostol or Rudin
Anyone who wants to understand WHY calculus works

📌 PREREQUISITES:
Basic understanding of sets and functions
Familiarity with rational and real numbers
Parts 1 & 2 of this series (recommended)

═══════════════════════════════════════

🔔 Subscribe for more rigorous mathematics made visual!

#RealAnalysis #Mathematics #Apostol #Supremum #CompletenessAxiom