1. Let a be a positive real number and n be a natural number. Define S = {x E R: x >0 and " < a}. (i) Prove that S is nonempty and bounded above. Let a = sup S. (ii) Assuming that a" < a, show that for some natural number m1, the inequality (a +1/m1)" < a holds. Why does this show that a" > a? (iii) Assuming that a" > a, show that for some natural number m2, the inequality (a – 1/m2)" > a holds. Why does this show that a" < a? (iv) Conclude from (ii) and (iii) above the existence of a positive nth root of a. Is such positive nth root of a unique? If yes then prove your claim, otherwise give a counterexample.

1. Let a be a positive real number and n be a natural number. Define S = {x E R: x >0 and " < a}. (i) Prove that S is nonempty and bounded above. Let a = sup S. (ii) Assuming that a" < a, show that for some natural number m1, the inequality (a +1/m1)" < a holds. Why does this show that a" > a? (iii) Assuming that a" > a, show that for some natural number m2, the inequality (a – 1/m2)" > a holds. Why does this show that a" < a? (iv) Conclude from (ii) and (iii) above the existence of a positive nth root of a. Is such positive nth root of a unique? If yes then prove your claim, otherwise give a counterexample.

Name: 1. Let a be a positive real number and n be a natural number. Define
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Description: 1. Let a be a positive real number and n be a natural number. Define S = {x €R:x >0 and x" < a}.(i) Prove that S is nonempty and bounded above. Let a = sup S.(ii) Assuming that a" < a, show that for some natural number m1, the inequality (a + 1/m1)"

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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1. Let a be a positive real number and n be a natural number. Define S = {x €R:x >0 and x" < a}.
(i) Prove that S is nonempty and bounded above. Let a = sup S.
(ii) Assuming that a" < a, show that for some natural number m1, the inequality (a + 1/m1)" < a
holds. Why does this show that a" > a?
(iii) Assuming that a" > a, show that for some natural number m2, the inequality (a – 1/m2)" > a
holds. Why does this show that a" < a?
(iv) Conclude from (ii) and (iii) above the existence of a positive nth root of a. Is such positive nth
root of a unique? If yes then prove your claim, otherwise give a counterexample.

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