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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.

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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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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