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Let a be a positive real number and n be a natural number. Define S = {r €R:r20 and a" < 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 m, the inequality (a + 1/m1)" < a holds. Why does this show that a"z 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"

Let a be a positive real number and n be a natural number. Define S = {r €R:r20 and a" < 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 m, the inequality (a + 1/m1)" < a holds. Why does this show that a"z 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"

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
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. Let a be a positive real number andn be a natural number. Define S = {a €R:r 20 and a" < 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" 2 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 o" < 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.
Transcribed Image Text:. Let a be a positive real number andn be a natural number. Define S = {a €R:r 20 and a" < 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" 2 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 o" < 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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