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).))

Algebra & Trigonometry with Analytic Geometry

13th Edition

ISBN:9781133382119

Author:Swokowski

Publisher:Swokowski

Chapter3: Functions And Graphs

Section3.2: Graphs Of Equations

Problem 68E

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