X 42. If x, y, a, and b are greater than zero and 43 Let A = {r: ris a rational numb y < , prove that b X x + a ×1> V y + b V b
X 42. If x, y, a, and b are greater than zero and 43 Let A = {r: ris a rational numb y < , prove that b X x + a ×1> V y + b V b
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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#42 please
![Certainly! Here's a transcription suitable for an educational website:
---
### Mathematical Concepts and Projects
#### Problem 41
If \( 0 < a < b \), prove that \( 0 < a^2 < b^2 \) and \( 0 < \sqrt{a} < \sqrt{b} \).
#### Problem 42
If \( x, y, a, \) and \( b \) are greater than zero and \( \frac{x}{y} < \frac{a}{b} \), prove that \( \frac{x + a}{y + b} < \frac{a}{b} \).
#### Problem 43
Let \( A = \{ r : r \text{ is a rational number and } r^2 < 2 \} \). Prove that \( A \) has no largest member. *(Hint: If \( r^2 < 2 \), and \( r > 0 \), choose a rational number \( \delta \) such that \( 0 < \delta < 1 \) and \( \delta < \frac{2 - r^2}{2r + 1} \). Show that \( (r + \delta)^2 < 2 \).*
#### Problem 44
If \( x = \sup S \), show that, for each \( \epsilon > 0 \), there is an \( s \in S \) such that \( x - \epsilon < s \leq x \).
#### Problem 45
If \( y = \inf S \), show that, for each \( \epsilon > 0 \), there is an \( s \in S \) such that \( y \leq s < y + \epsilon \).
### PROJECT 0.1
The purpose of this project is to show that the open interval \( (0, 1) \) is equivalent to the closed interval \( [0, 1] \). In the process we will discover that both intervals are equivalent to \( [0, 1) \) and \( (0, 1] \). It is then easy to generalize to any interval \([a, b]\) with \( a < b \).
---
This transcription includes the mathematical problems and the introduction to Project 0.1 as found in the image.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fd597ffd2-5c4b-4c2e-8332-77ce1607dac1%2F19a0ffdb-22c3-4e89-90f1-febab758cfd4%2Fz5slpms_processed.jpeg&w=3840&q=75)
Transcribed Image Text:Certainly! Here's a transcription suitable for an educational website:
---
### Mathematical Concepts and Projects
#### Problem 41
If \( 0 < a < b \), prove that \( 0 < a^2 < b^2 \) and \( 0 < \sqrt{a} < \sqrt{b} \).
#### Problem 42
If \( x, y, a, \) and \( b \) are greater than zero and \( \frac{x}{y} < \frac{a}{b} \), prove that \( \frac{x + a}{y + b} < \frac{a}{b} \).
#### Problem 43
Let \( A = \{ r : r \text{ is a rational number and } r^2 < 2 \} \). Prove that \( A \) has no largest member. *(Hint: If \( r^2 < 2 \), and \( r > 0 \), choose a rational number \( \delta \) such that \( 0 < \delta < 1 \) and \( \delta < \frac{2 - r^2}{2r + 1} \). Show that \( (r + \delta)^2 < 2 \).*
#### Problem 44
If \( x = \sup S \), show that, for each \( \epsilon > 0 \), there is an \( s \in S \) such that \( x - \epsilon < s \leq x \).
#### Problem 45
If \( y = \inf S \), show that, for each \( \epsilon > 0 \), there is an \( s \in S \) such that \( y \leq s < y + \epsilon \).
### PROJECT 0.1
The purpose of this project is to show that the open interval \( (0, 1) \) is equivalent to the closed interval \( [0, 1] \). In the process we will discover that both intervals are equivalent to \( [0, 1) \) and \( (0, 1] \). It is then easy to generalize to any interval \([a, b]\) with \( a < b \).
---
This transcription includes the mathematical problems and the introduction to Project 0.1 as found in the image.
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