(c) Let us assume the facts from part (b) for all a, b e R, i.e. as < bs whenever a < b. Let SCR be bounded above and non-empty. Define Š = {x € R[x* € S}. Prove that sup S sup S3 using the definition of the least upper bound. %3D

Advanced Engineering Mathematics
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ISBN:9780470458365
Author:Erwin Kreyszig
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Chapter2: Second-order Linear Odes
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Use the info in b to solve for c. 

(b) Define x = aš as the real number satisfying the equality (x · x) · x = a. For a > 0, you may
assume as true that as > 0. Use only this definition and the axiomatic definition of the real
numbers, to prove that
a < b3
whenever 0 <a< b. You may assume that for every a E R, there is a number as in R. Justify
each step and use only one axiom per step.
(c) Let us assume the facts from part (b) for all a, b e R, i.e. as < b3 whenever a < b. Let
SCR be bounded above and non-empty. Define
Š = {r € R]r³ € S}.
Prove that
sup
Š =
= sup Si using the definition of the least upper bound.
Transcribed Image Text:(b) Define x = aš as the real number satisfying the equality (x · x) · x = a. For a > 0, you may assume as true that as > 0. Use only this definition and the axiomatic definition of the real numbers, to prove that a < b3 whenever 0 <a< b. You may assume that for every a E R, there is a number as in R. Justify each step and use only one axiom per step. (c) Let us assume the facts from part (b) for all a, b e R, i.e. as < b3 whenever a < b. Let SCR be bounded above and non-empty. Define Š = {r € R]r³ € S}. Prove that sup Š = = sup Si using the definition of the least upper bound.
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