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

Let S ⊆ R and M = sup S which is the least upper bound of S. We are given that S is bounded, so sup…

Answered: Let S ⊆ R, and recall M = supS is the…

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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Let S ⊆ R, and recall M = supS is the least upper bound of S. Suppose that S is bounded, so supS ∈ R. (If S was not bounded, then we set supS = ∞). Show that for every ε > 0, there exists x ∈ S such that |x−M| < ε and that there exists a sequence (x_n) ^∞_n=1 from S such that for each n ∈ N,

|x_n− M| < 1/2ⁿ.

Define M to be the supremum of S. R and N are the real numbers and natural numbers, respectively.

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