Let S be a non-empty set of real numbers which is bounded above, andsuppose that x = sup(S).(a) Prove that for all € > 0 there exists s ES so that s = (x − €, x].(b) Suppose now that x S. Prove that for every e > 0 the set {s ES |se (x - e, x]} is infinite.

We will use definition of least upper bound or supremum .

Answered: Let S be a non-empty set of real…

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 be a non-empty set of real numbers which is bounded above, and suppose that x = sup(S). - (a) Prove that for all € > 0 there exists s € S so that s ≤ (x − €, x].