(a) Triangle Inequality Let a and b be real numbers. Then (b) Reverse Triangle Inequality Let a and b be real numbers. Then lal - IM S la -M CONVERGENCE OF SEQUENCES AND SERIES 80 Problem 65. (a) Prove Lemma 0/ Hint: For the Reverse Triangle Inequality, consider Jal - la -b+./
(a) Triangle Inequality Let a and b be real numbers. Then (b) Reverse Triangle Inequality Let a and b be real numbers. Then lal - IM S la -M CONVERGENCE OF SEQUENCES AND SERIES 80 Problem 65. (a) Prove Lemma 0/ Hint: For the Reverse Triangle Inequality, consider Jal - la -b+./
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
100%
#65 part a
![Certainly! Here's a transcription suitable for an educational website:
---
### Convergence of Sequences and Series
**Problem 64.** Let \( c_n = c \) (i.e., \( c, c, c, \ldots \)) be a constant sequence. Show that \(\lim_{n \to \infty} c_n = c\).
In proving the familiar limit theorems, the following will prove to be a very useful tool.
**Lemma 1.**
(a) **Triangle Inequality**: Let \( a \) and \( b \) be real numbers. Then
\[ |a + b| \leq |a| + |b|. \]
(b) **Reverse Triangle Inequality**: Let \( a \) and \( b \) be real numbers. Then
\[ ||a| - |b|| \leq |a - b|. \]
---
**Problem 65.**
(a) *Prove Lemma 1.* [Hint: For the Reverse Triangle Inequality, consider \( |a| = |a - b + b|. \)]
(b) *Show* \( ||a| - |b|| \leq |a - b| \). [Hint: You want to show \( |a| - |b| \leq |a - b| \) and \(-(|a| - |b|) \leq |a - b|. \)]
**Theorem 7.** If \(\lim_{n \to \infty} a_n = a\) and \(\lim_{n \to \infty} b_n = b\), then \(\lim_{n \to \infty} (a_n + b_n) = a + b\).
We will often informally state this theorem as "the limit of a sum is the sum of the limits". However, to be absolutely precise, what this says is that if we already know that two sequences converge, then the sequence formed by summing the corresponding terms of those two sequences will converge and, in fact, converge to the sum of those individual limits. We’ll provide the scrapwork for the proof of this and leave the formal write-up as an exercise. Note the use of the triangle inequality in the proof.
**SCRAPWORK:** If we let \(\varepsilon > 0\), then we want \( N \) so that](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fc074fc7e-4e48-46f1-bfc4-aee6ba4e6262%2F209892a9-48d2-4b73-8e91-83df3ee63b8e%2Fmikg6k_processed.jpeg&w=3840&q=75)
Transcribed Image Text:Certainly! Here's a transcription suitable for an educational website:
---
### Convergence of Sequences and Series
**Problem 64.** Let \( c_n = c \) (i.e., \( c, c, c, \ldots \)) be a constant sequence. Show that \(\lim_{n \to \infty} c_n = c\).
In proving the familiar limit theorems, the following will prove to be a very useful tool.
**Lemma 1.**
(a) **Triangle Inequality**: Let \( a \) and \( b \) be real numbers. Then
\[ |a + b| \leq |a| + |b|. \]
(b) **Reverse Triangle Inequality**: Let \( a \) and \( b \) be real numbers. Then
\[ ||a| - |b|| \leq |a - b|. \]
---
**Problem 65.**
(a) *Prove Lemma 1.* [Hint: For the Reverse Triangle Inequality, consider \( |a| = |a - b + b|. \)]
(b) *Show* \( ||a| - |b|| \leq |a - b| \). [Hint: You want to show \( |a| - |b| \leq |a - b| \) and \(-(|a| - |b|) \leq |a - b|. \)]
**Theorem 7.** If \(\lim_{n \to \infty} a_n = a\) and \(\lim_{n \to \infty} b_n = b\), then \(\lim_{n \to \infty} (a_n + b_n) = a + b\).
We will often informally state this theorem as "the limit of a sum is the sum of the limits". However, to be absolutely precise, what this says is that if we already know that two sequences converge, then the sequence formed by summing the corresponding terms of those two sequences will converge and, in fact, converge to the sum of those individual limits. We’ll provide the scrapwork for the proof of this and leave the formal write-up as an exercise. Note the use of the triangle inequality in the proof.
**SCRAPWORK:** If we let \(\varepsilon > 0\), then we want \( N \) so that
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
Step by step
Solved in 2 steps with 1 images
Similar questions
Recommended textbooks for you
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Mathematics For Machine Technology
Advanced Math
ISBN:
9781337798310
Author:
Peterson, John.
Publisher:
Cengage Learning,
