Theorem: Let A = {x € Q : x < √2}. Then sup(A) = √√2. Proof: For all x E A we have x We must now show that √2 is the the rationals in the reals, there exists a y E least greatest which shows that z is irrational number rational number √2. This shows that √2 is > upper bound. Consider z E R with z y such that z least upper bound y and y for A. We conclude that √2 is the A for A. an upper bound not an upper bound √2. By the density of √2. Then <

Theorem: Let A = {x € Q : x < √2}. Then sup(A) = √√2. Proof: For all x E A we have x We must now show that √2 is the the rationals in the reals, there exists a y E least greatest which shows that z is irrational number rational number √2. This shows that √2 is > upper bound. Consider z E R with z y such that z least upper bound y and y for A. We conclude that √2 is the A for A. an upper bound not an upper bound √2. By the density of √2. Then <

Algebra & Trigonometry with Analytic Geometry

13th Edition

ISBN:9781133382119

Author:Swokowski

Publisher:Swokowski

Chapter1: Fundamental Concepts Of Algebra

Section1.1: Real Numbers

Problem 36E

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