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3. Let d(x, y) := |2 - 2| for x, y E R. . is a metric on R. Show that (R, d) is not a complete metric space. (Hint: Consider the sequence (xn) := (n), and use the fact that 2-n →0 as n → 00.) Then (Comment: This gives an example of a metric on R with respect to which R is not complete. (Recall that R is complete with respect to the (usual) Euclidean metric (by Lem. 4.20).))

3. Let d(x, y) := |2 - 2| for x, y E R. . is a metric on R. Show that (R, d) is not a complete metric space. (Hint: Consider the sequence (xn) := (n), and use the fact that 2-n →0 as n → 00.) Then (Comment: This gives an example of a metric on R with respect to which R is not complete. (Recall that R is complete with respect to the (usual) Euclidean metric (by Lem. 4.20).))

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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3. Let d(x, y) |2 -2 for x, y ER. Then is a metric on R. Show that (R, d) is not a complete metric space. (Hint: Consider the sequence (Tn) (Comment: This gives an example of a metric on R with respect to which R is not complete. (Recall that R is complete with respect to the (usual) Euclidean metric (by Lem. 4.20).)) (n), and use the fact that 2n → 0 as n → 00.)
Transcribed Image Text:3. Let d(x, y) |2 -2 for x, y ER. Then is a metric on R. Show that (R, d) is not a complete metric space. (Hint: Consider the sequence (Tn) (Comment: This gives an example of a metric on R with respect to which R is not complete. (Recall that R is complete with respect to the (usual) Euclidean metric (by Lem. 4.20).)) (n), and use the fact that 2n → 0 as n → 00.)
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