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.

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.

Name: This is an easy question of Real analysis 1. Let a be a positive real
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Description: This is an easy question of Real analysis 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 natu

Advanced Engineering Mathematics

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ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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This is an easy question of Real analysis

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.

Branch of mathematical analysis that studies real numbers, sequences, and series of real numbers and real functions. The concepts of real analysis underpin calculus and its application to it. It also includes limits, convergence, continuity, and measure theory.

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