(iv) If a sequence {n} in X converges, it converges to a unique point. (iv) Every convergent sequence in X is a Cauchy sequence. (v) A subset Y of a complete metric space is (X, d) is complete if and only if Y is closed and Y has the metric inherited from (X, d). (vi) Show that the subset (0, 1] of the set of reals R with usual metric is not complete.
(iv) If a sequence {n} in X converges, it converges to a unique point. (iv) Every convergent sequence in X is a Cauchy sequence. (v) A subset Y of a complete metric space is (X, d) is complete if and only if Y is closed and Y has the metric inherited from (X, d). (vi) Show that the subset (0, 1] of the set of reals R with usual metric is not complete.
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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Question
Please solve 3iv, 3iv, 3v, 3vi
![3. Prove each of the following for a metric space (X, d):
(i) Union of two closed sets is closed.
(ii) A set AC X is closed if and only if for each x EX-A, there is
an open set U containing a such that UnA = 0.
(iii) For A CX, xe clA, if and only if there is a sequence {n} A
such that {n} →→x.
(iv) If a sequence {n} in X converges, it converges to a unique point.
(iv) Every convergent sequence in X is a Cauchy sequence.
(v) A subset Y of a complete metric space is (X, d) is complete if
and only if Y is closed and Y has the metric inherited from (X, d).
(vi) Show that the subset (0, 1] of the set of reals R with usual metric
is not complete.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Faeb81122-6fbf-4c63-aaf5-71689ff4fea7%2F04edae32-c4ee-4ba0-8b67-42e85c9fba75%2Fd8d3zmr_processed.jpeg&w=3840&q=75)
Transcribed Image Text:3. Prove each of the following for a metric space (X, d):
(i) Union of two closed sets is closed.
(ii) A set AC X is closed if and only if for each x EX-A, there is
an open set U containing a such that UnA = 0.
(iii) For A CX, xe clA, if and only if there is a sequence {n} A
such that {n} →→x.
(iv) If a sequence {n} in X converges, it converges to a unique point.
(iv) Every convergent sequence in X is a Cauchy sequence.
(v) A subset Y of a complete metric space is (X, d) is complete if
and only if Y is closed and Y has the metric inherited from (X, d).
(vi) Show that the subset (0, 1] of the set of reals R with usual metric
is not complete.
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