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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 <

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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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 V the rationals in the reals, there exists a ує least greatest which shows that z is irrational number rational number √2. This shows that √2 is upper bound. Consider z ER 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 <
Transcribed Image Text: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 V the rationals in the reals, there exists a ує least greatest which shows that z is irrational number rational number √2. This shows that √2 is upper bound. Consider z ER 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 <
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Step 1: Prove the theorem:

Let A={xelement ofQ:x<√2}. Then sup(A)=√2.

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