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3. Suppose that {nk ke N} is a strictly increasing sequence of natural numbers and {n ne N} is a sequence in a metric space X. Then we say the sequence {nk EN} is a subsequence of {n}. Suppose that {rn E N} is a Cauchy sequence in the metric space (X, d) which has a convergent subsequence. Prove that {n} converges.

3. Suppose that {nk ke N} is a strictly increasing sequence of natural numbers and {n ne N} is a sequence in a metric space X. Then we say the sequence {nk EN} is a subsequence of {n}. Suppose that {rn E N} is a Cauchy sequence in the metric space (X, d) which has a convergent subsequence. Prove that {n} converges.

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: (i) The set R of rational numbers is first category in the set of Reals (ii) The set of rational numbers cannot be written as Q = EN Un where {Un ne N} is a sequence of open subsets of R, the set of all Reals. 2. Prove that the set 2 of integers with restriction of usual metric on R is a complete metric space, yet it is the union of countably many singletons. Explain why this does not contradict the Baire category theorem? 3. Suppose that {nk ke N} is a strictly increasing sequence of natural numbers and {xn n E N} is a sequence in a metric space X. Then we say the sequence {nk N} is a subsequence of {n}. Suppose that {n E N} is a Cauchy sequence in the metric space (X, d) which has a convergent subsequence. Prove that {n} converges. 4. Suppose that X, Y and Z are metric spaces and F X Y and g Y Z are continuous functions. prove that go f X→ Z is continuous. : 1
Transcribed Image Text:1. Prove each of the following: (i) The set R of rational numbers is first category in the set of Reals (ii) The set of rational numbers cannot be written as Q = EN Un where {Un ne N} is a sequence of open subsets of R, the set of all Reals. 2. Prove that the set 2 of integers with restriction of usual metric on R is a complete metric space, yet it is the union of countably many singletons. Explain why this does not contradict the Baire category theorem? 3. Suppose that {nk ke N} is a strictly increasing sequence of natural numbers and {xn n E N} is a sequence in a metric space X. Then we say the sequence {nk N} is a subsequence of {n}. Suppose that {n E N} is a Cauchy sequence in the metric space (X, d) which has a convergent subsequence. Prove that {n} converges. 4. Suppose that X, Y and Z are metric spaces and F X Y and g Y Z are continuous functions. prove that go f X→ Z is continuous. : 1
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