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

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.

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

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🎯 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)

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