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2. Let X be the space of all ordered n-tuples x=(₁,,n) of real numbers and d(x, y) = max |; - ni

2. Let X be the space of all ordered n-tuples x=(₁,,n) of real numbers and d(x, y) = max |; - ni

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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2. Let X be the space of all ordered n-tuples x=(₁,,n) of real numbers and d(x, y) = max ₁-nil where y = (n). Show that (X, d) is complete. 3. Let Mcl be the subspace consisting of all sequences x = (5) with at most finitely many nonzero terms. Find a Cauchy sequence in M which does not converge in M, so that M is not complete. 4. Show that M in Prob. 3 is not complete by applying Theorem 1.4-7. 5. Show that the set X of all integers with metric d defined by d(m, n) |m-n is a complete metric space.
Transcribed Image Text:2. Let X be the space of all ordered n-tuples x=(₁,,n) of real numbers and d(x, y) = max ₁-nil where y = (n). Show that (X, d) is complete. 3. Let Mcl be the subspace consisting of all sequences x = (5) with at most finitely many nonzero terms. Find a Cauchy sequence in M which does not converge in M, so that M is not complete. 4. Show that M in Prob. 3 is not complete by applying Theorem 1.4-7. 5. Show that the set X of all integers with metric d defined by d(m, n) |m-n is a complete metric space.
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