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Unformatted text preview: (c) A sequence with subsequences converging to each of the integers. (d) A sequence ( a n ) so that ( a n ) converges, but ∞ X n =1 a n diverges. 7. Prove or give a counter-example. (a) If ( a n ) converges to zero, then for any b , ( a n + b ) converges to b . (b) If ( a n ) is monotone increasing, then it is bounded. (c) If b k is a sequence of positive integers, then a n = n X k =1 b k is a sequence monotone increasing. 8. State the following theorems and properties and their negations. (a) Monotone Convergence Theorem. (b) Archimedean property. (c) Nested Interval Property. (d) Cauchy Theorem. 9. State the following deﬁnitions and their negations. Give an example, and a non-example for each one. (a) A Cauchy sequence. (b) The supremum of a set. (c) A nested interval. (d) A convergent series. 2...
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- Fall '08
- Mathematical analysis, Cauchy sequence, Dominated convergence theorem, Archimedean