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8. Let SR, and recall M= supS is the least upper bound of S. Suppose that S is bounded, so supS E R. (If S was not bounded, then we set supS = ∞o). Show that for every > 0, there exists x € S such that |x-M| <ε and that there exists a sequence (x₁) n=1 from S such that for each n E N, xn-M<1/2". Define M to be the supremum of S. R and N are the real numbers and natural numbers, respectively.

8. Let SR, and recall M= supS is the least upper bound of S. Suppose that S is bounded, so supS E R. (If S was not bounded, then we set supS = ∞o). Show that for every > 0, there exists x € S such that |x-M| <ε and that there exists a sequence (x₁) n=1 from S such that for each n E N, xn-M<1/2". Define M to be the supremum of S. R and N are the real numbers and natural numbers, respectively.

Algebra & Trigonometry with Analytic Geometry
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter10: Sequences, Series, And Probability
Section10.1: Infinite Sequences And Summation Notation
Problem 74E
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8. Let SCR, and recall M = supS is the least upper bound of S. Suppose that S is bounded, so supS ER. (If S was not bounded, then we set sups = 00). Show that for every ɛ > 0, there exists x ES such that |x-M| 00 <ɛ and that there exists a sequence (x,) n=1 from S such that for each n EN, |Xn – M| < 1/2". Define M to be the supremum of S. R and N are the real numbers and natural numbers, respectively.
Transcribed Image Text:8. Let SCR, and recall M = supS is the least upper bound of S. Suppose that S is bounded, so supS ER. (If S was not bounded, then we set sups = 00). Show that for every ɛ > 0, there exists x ES such that |x-M| 00 <ɛ and that there exists a sequence (x,) n=1 from S such that for each n EN, |Xn – M| < 1/2". Define M to be the supremum of S. R and N are the real numbers and natural numbers, respectively.
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