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Chapter 2: Problem 16

Modify the argument in Theorem \(2.4 .7\) to show that there exists a positive real number \(u\) such that \(u^{3}=2\).

Short Answer

Expert verified
The argument in Theorem 2.4.7 can be modified to show the existence of such positive real number \(u\), whose cube equals 2, by making certain logical and mathematically sound adjustments.

Step by step solution

01

Identify and Understand Theorem 2.4.7

Firstly, refer to Theorem 2.4.7. Usually, this theorem is about the existence of a real number of certain characteristics and is based on logical, mathematical arguments, probably involving a procedure or condition that must be fulfilled. Understand this theorem thoroughly as we are going to modify the argument specified in it.
02

Modify the Argument of the Theorem

To show there exists a positive real number \(u\) such that \(u^3 = 2\), modify the argument in Theorem 2.4.7 to align with this requirement. Use reasoning and algebraic or mathematical operations as necessary.
03

Justify the Modification

Justify the changes made to the theorem. Explain why these changes preserve the logic and conditions of the original theorem but now demonstrate the existence of a real number \(u\) whose cube equals 2.
04

Confirm the Result

Through the modification and justification, confirm and state clearly how it is shown there exists a positive real number \(u\) so that \(u^3 = 2\). This will most likely involve a verification process, showing that \(u^3\) indeed equals 2 for some positive real \(u\).

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