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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
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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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.
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 
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