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P.1. Problem Set (Easy-Medium)

# P.1. Problem Set (Easy-Medium) - #1 The goal of this...

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#1 The goal of this problem is to ensure mastery of the basic concepts concerning sequences. Recall Definition: A sequence converges to a real number L if for every , there is some positive integer N so that for all , . Exercise: Verify, using the definition of limit of a sequence, that a. b. c. Exercise: If converges to A , prove that converges to . Show by example that the converse need not be true. Exercise: Give an example for each of the following or prove that no such example exists: a. A bounded sequence that does not converge b. A bounded sequence that does converge c. A sequence that converges to a real number L but is not bounded d. A monotone sequence that is Cauchy e. Sequences and which do not converge, but converges f. Repeat part e . replacing with

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#2 The goal of this problem is to ensure mastery of the basic concepts concerning limits of functions. Recall Definition: Let be some function and let p be an accumulation point of D . Then f has limit L at p if for every , there is such that for all satisfying , it follows that .
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P.1. Problem Set (Easy-Medium) - #1 The goal of this...

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