Prove that if a, b, and c are positive real numbers with ab = c, then a < Vc or b< Vc.
Prove that if a, b, and c are positive real numbers with ab = c, then a < Vc or b< Vc.
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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question 6。mathematical reasoning. Show steps，probably by p or q is equivalent to not p =》q
![1. Prove by contradiction that \(6n + 5\) is odd for all integers \(n\).
2. Prove that for all integers \(n\), if \(3n + 5\) is even then \(n\) is odd. (Hint: prove the contrapositive)
3. Prove that
\[
|x + y| \leq |x| + |y|
\]
for all real numbers \(x\) and \(y\).
4. Prove that there does not exist a smallest positive real number. (In other words, prove that there does not exist a positive real number \(x\) such that \(x < y\) for all positive real numbers \(y\)).
5. Recall that an irrational number is a real number which is not rational. Prove that if \(x\) is rational and \(y\) is irrational, then \(x + y\) is irrational. You may use the fact that the rational numbers are closed under addition - if \(a\) and \(b\) are rational numbers, then \(a + b\) is rational as well.
6. Prove that if \(a\), \(b\), and \(c\) are positive real numbers with \(ab = c\), then \(a \leq \sqrt{c}\) or \(b \leq \sqrt{c}\).
7. Prove that
\[
\sum_{i=1}^n i^3 = \frac{1}{4}n^2(n+1)^2
\]
for all positive integers \(n\).
8. Prove that
\[
\sum_{i=1}^n i \cdot i! = (n+1)! - 1
\]
for all positive integers \(n\).
9. Prove that
\[
2^n < n!
\]
for all positive integers \(n\) such that \(n \geq 4\).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F67a79aa2-f715-406c-8bee-8178252bb86d%2F05dc51c4-b888-411b-9c13-7467ba50644f%2F19j46tq_processed.jpeg&w=3840&q=75)
Transcribed Image Text:1. Prove by contradiction that \(6n + 5\) is odd for all integers \(n\).
2. Prove that for all integers \(n\), if \(3n + 5\) is even then \(n\) is odd. (Hint: prove the contrapositive)
3. Prove that
\[
|x + y| \leq |x| + |y|
\]
for all real numbers \(x\) and \(y\).
4. Prove that there does not exist a smallest positive real number. (In other words, prove that there does not exist a positive real number \(x\) such that \(x < y\) for all positive real numbers \(y\)).
5. Recall that an irrational number is a real number which is not rational. Prove that if \(x\) is rational and \(y\) is irrational, then \(x + y\) is irrational. You may use the fact that the rational numbers are closed under addition - if \(a\) and \(b\) are rational numbers, then \(a + b\) is rational as well.
6. Prove that if \(a\), \(b\), and \(c\) are positive real numbers with \(ab = c\), then \(a \leq \sqrt{c}\) or \(b \leq \sqrt{c}\).
7. Prove that
\[
\sum_{i=1}^n i^3 = \frac{1}{4}n^2(n+1)^2
\]
for all positive integers \(n\).
8. Prove that
\[
\sum_{i=1}^n i \cdot i! = (n+1)! - 1
\]
for all positive integers \(n\).
9. Prove that
\[
2^n < n!
\]
for all positive integers \(n\) such that \(n \geq 4\).
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