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3. Define each of the following concepts for a metric space (X, d): (i) A sequence converges in a metric space; (ii) A Cauchy sequence in X; (iii) A complete metric space; (iv) Diameter of a subset A of X

3. Define each of the following concepts for a metric space (X, d): (i) A sequence converges in a metric space; (ii) A Cauchy sequence in X; (iii) A complete metric space; (iv) Diameter of a subset A of X

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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1. Prove each of the following for a metric space (X, d): (i) A set ACX is closed if and only if for each x E X-A, there is an open set U containing x such that UnA = 0. (ii) For A CX, x E clA, if and only if there is a sequence {n} A such that {n} →x. (iii) 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 X is complete if and only if Y is complete and Y has the metric inherited from (X, d). 2. State and prove the Cantor's Intersection Theorem. 3. Define each of the following concepts for a metric space (X, d): (i) A sequence converges in a metric space; (ii) A Cauchy sequence in X; (iii) A complete metric space; (iv) Diameter of a subset A of X
Transcribed Image Text:1. Prove each of the following for a metric space (X, d): (i) A set ACX is closed if and only if for each x E X-A, there is an open set U containing x such that UnA = 0. (ii) For A CX, x E clA, if and only if there is a sequence {n} A such that {n} →x. (iii) 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 X is complete if and only if Y is complete and Y has the metric inherited from (X, d). 2. State and prove the Cantor's Intersection Theorem. 3. Define each of the following concepts for a metric space (X, d): (i) A sequence converges in a metric space; (ii) A Cauchy sequence in X; (iii) A complete metric space; (iv) Diameter of a subset A of X
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