Let A, B C R. We define the distance between A and B by d(A, B) = inf{|x - y| : x € A, y E B}. Show that if A is compact and B is closed, and d(A, B) = 0, then ANB÷Ø.
Let A, B C R. We define the distance between A and B by d(A, B) = inf{|x - y| : x € A, y E B}. Show that if A is compact and B is closed, and d(A, B) = 0, then ANB÷Ø.
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
How do you solve the first bullet point?
Transcribed Image Text:Let A, B C R. We define the distance between A and B by
d(A, B) = inf{|x - y| : x E A, y E B}.
Show that if A is compact and B is closed, and d(A, B) = 0, then ANB#Ø.
• Give examples of d(A, B) = 0, ANB=Ø when
o A and Bare merely closed;
o A and Bare open and bounded.
Expert Solution
Step 1 Points to remember
- A is compact impies that every sequence in A has a convergent subsequence converging to a point in A
- A is closed implies every limit point of A belongs to A
- Difference of two convergent sequence is convergent
- Every subsequence of a convergent sequence converging to the same limit
Step by step
Solved in 3 steps with 2 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,
