(b) Define x = a³ 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 < b% 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.
Unitary Method
The word “unitary” comes from the word “unit”, which means a single and complete entity. In this method, we find the value of a unit product from the given number of products, and then we solve for the other number of products.
Speed, Time, and Distance
Imagine you and 3 of your friends are planning to go to the playground at 6 in the evening. Your house is one mile away from the playground and one of your friends named Jim must start at 5 pm to reach the playground by walk. The other two friends are 3 miles away.
Profit and Loss
The amount earned or lost on the sale of one or more items is referred to as the profit or loss on that item.
Units and Measurements
Measurements and comparisons are the foundation of science and engineering. We, therefore, need rules that tell us how things are measured and compared. For these measurements and comparisons, we perform certain experiments, and we will need the experiments to set up the devices.
I was wondering how to solve b,c,d, and e. (All of them)
![(b) Define x = a3 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
aš < b3
whenever 0 <a<b. You may assume that for every a E R, there is a number aš 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. aš < b3 whenever a < b. Let
SCR be bounded above and non-empty. Define
Š = {x € R]x* € S}.
Prove that sup
= sup
using the definition of the least upper bound.
(d) Consider the sequence {an} C Q given by
п 3 2, 4, 7, 17
n+1
2n
otherwise
Prove the sequence is Cauchy from the definition of a Cauchy sequence.
(e) In the context of the Cauchy sequence construction of the real numbers, find the inverses
-{an} and {an}-1 for the sequence of part (d). No partial credit. Only give the final answer
for this part.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fa9c6cd38-7675-4fcb-acad-b439c4080f5d%2F75327d2d-84bf-4b55-9d70-953a81aa7d75%2Fgrm7d1_processed.png&w=3840&q=75)
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