Problem Set
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Chapter
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Problems
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| 01 |
[NA] |
Look over the course webpage. In particular, read through the course syllabus, get accustomed to requesting a course calendar, and look over the interface for getting homework assignments.
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∗1 |
Prove the assertion: If (X, d ) is a metric space and Y is a subset of X, then (Y, d ) is a metric space.
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∗2 |
Prove the assertion: If (xn ) is a Cauchy sequence in a metric space (X, d ), then the set {xn | n = 1, 2, ...} is bounded.
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∗3 |
Prove the assertion: Suppose (X, d ) is a complete metric space, and (xn ) is a sequence in X. Then (xn ) is convergent if and only if it is Cauchy.
Note: An “if and only if” statement always makes two assertions. Which, if either one, of the two assertions made here still holds if X is not complete?
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2 |
11 |