Prove B = {ne N} U {0} is compact.
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
Using the Heine-Borel Theorem
![**Proof of Compactness for Set \( B \)**
**Problem:**
Prove that the set \( B = \left\{\frac{1}{n} : n \in \mathbb{N}\right\} \cup \{0\} \) is compact.
**Explanation:**
- **Set Definition:** The set \( B \) consists of the reciprocals of natural numbers together with the element 0. This set can be expressed as:
\[
B = \left\{\frac{1}{1}, \frac{1}{2}, \frac{1}{3}, \ldots \right\} \cup \{0\}
\]
- **Compactness in \(\mathbb{R}\):** A subset of real numbers is compact if it is both closed and bounded.
- **Boundedness:**
- The set \( B \subseteq [0, 1] \) because for any \( n \in \mathbb{N} \), \( 0 \leq \frac{1}{n} \leq 1 \).
- Thus, \( B \) is bounded.
- **Closedness:**
- To show \( B \) is closed, consider the limit point.
- As \( n \to \infty \), \(\frac{1}{n} \to 0\).
- The set includes the point 0, thus containing its limit point.
- Therefore, \( B \) is closed in \(\mathbb{R}\).
Since \( B \) is both closed and bounded, \( B \) is compact by the Heine–Borel theorem.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F545e4110-447f-42e0-86d8-f710095b2322%2Fd0e389c4-7251-407f-8368-c3315ef44e5e%2Fh8xhifb_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Proof of Compactness for Set \( B \)**
**Problem:**
Prove that the set \( B = \left\{\frac{1}{n} : n \in \mathbb{N}\right\} \cup \{0\} \) is compact.
**Explanation:**
- **Set Definition:** The set \( B \) consists of the reciprocals of natural numbers together with the element 0. This set can be expressed as:
\[
B = \left\{\frac{1}{1}, \frac{1}{2}, \frac{1}{3}, \ldots \right\} \cup \{0\}
\]
- **Compactness in \(\mathbb{R}\):** A subset of real numbers is compact if it is both closed and bounded.
- **Boundedness:**
- The set \( B \subseteq [0, 1] \) because for any \( n \in \mathbb{N} \), \( 0 \leq \frac{1}{n} \leq 1 \).
- Thus, \( B \) is bounded.
- **Closedness:**
- To show \( B \) is closed, consider the limit point.
- As \( n \to \infty \), \(\frac{1}{n} \to 0\).
- The set includes the point 0, thus containing its limit point.
- Therefore, \( B \) is closed in \(\mathbb{R}\).
Since \( B \) is both closed and bounded, \( B \) is compact by the Heine–Borel theorem.
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
This is a popular solution!
Trending now
This is a popular solution!
Step by step
Solved in 3 steps with 1 images
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,
