1. The metric subspace 1, 2] of the Euclidean metric space R is a complete metric space. a. True b. False 2. Every Cauchy sequence in the Euclidean metric space R", where n is a positive integer, is convergent. a. True b. False 3. Every subsequence of a Cauchy sequence is a Cauchy sequence. a. True b. False

1. The metric subspace 1, 2] of the Euclidean metric space R is a complete metric space. a. True b. False 2. Every Cauchy sequence in the Euclidean metric space R", where n is a positive integer, is convergent. a. True b. False 3. Every subsequence of a Cauchy sequence is a Cauchy sequence. a. True b. False

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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The given is on top please solve part 1 and 2
6. Which of the following statements is false in a metric space (X, d)? (a) Every Cauchy sequence is convergent. (b) Every open ball is an open set. (c) Every open set is a union of open balls. (d) Closed sets are complements of open sets. 7. Which of the following subsets of R is necessarily open under the discrete metric? (a) infinite sets (b) finite sets (c) intervals (d) all of the choices
Question 1 (1.1) Let {Y;}1
(1.2) Let Y = {Y,o} be a metric space. Suppose (pn) and (q,) are sequences in Y, where (9n) is a Cauchy sequence and o(pn; qn) → 0 as n → x. Show the following : (a) (Pn) is a Cauchy sequence in Y, (b) Pn + a if and only if q, → x.
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Please give a perfect answer
Suppose that a sequence xn in a Hilbert space H is convergent to some limit in x in H. a) What does this statement mean exactly? (Give an epsilon N definition). b) Show that xn is Cauchy. c) Use the conclusion in (b) to further conclude that H is closed.
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