Archimedean Property - Proof and Some Application | Real Analysis |The Order of Mathematics скачать в хорошем качестве

Archimedean Property - Proof and Some Application | Real Analysis |The Order of Mathematics

In this video, we provide a detailed explanation and formal proof of the Archimedean Property, a fundamental principle in Real Analysis. We begin by reviewing the Least Upper Bound (LUB) Property (Completeness Axiom) of real numbers and demonstrate how it is used to prove that the set of natural numbers (\mathbb{N}) is not bounded above in \mathbb{R}. Key Topics Covered: The LUB Property: Understanding upper bounds and the existence of the supremum for non-empty sets bounded above [00:22]. Formal Proof of Archimedean Property: Using contradiction to show that for any real number z, there exists a natural number n such that n greater than z [04:18]. Alternative Version: Proving that for any positive real numbers x and y, there exists a natural number n such that nx greater than y [07:00]. Reciprocal Property: Showing that for any x greater than 0, there exists an n such that x greater than 1/n [09:42]. Characterization of Zero: Proving that if x \ge 0 and x \le 1/n for all n \in \mathbb{N}, then x = 0 [10:39]. Application: How to use these properties to prove that two real numbers are equal (a = b) [12:33]. This tutorial is essential for students preparing for exams like CSIR NET, IIT JAM, or University-level Real Analysis courses. Keywords Real Analysis, Mathematics, Calculus, Higher Mathematics, Pure Mathematics, CSIR NET Mathematics, IIT JAM Maths, University Math. Archimedean Property, Archimedean Property Proof, Least Upper Bound Property, LUB Property, Completeness Axiom, Supremum Property, Infimum and Supremum, Bounded Sets, Real Number System, Sequence and Series.