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Unformatted text preview: #1 The goal of this problem is to ensure mastery of the basic concepts concerning sequences. Let Exercise: First, define what it means for a sequence to converge to a real number L . Do the same for what it means for a sequence to be Cauchy. Exercise: Give an proof for each of the following: a. b. c. ( Hint: Prove that for all positive integers n. ) d. ( Hint: Note that . Then insist . ) Exercise: Prove that each of the sequences above are also Cauchy. You may do so by proving if a sequence converges, then it must also be Cauchy. Exercise: A sequence is called pseudo-Cauchy if for every , there is some N so that for all , it follows that . Show that the sequence defined by is pseudo-Cauchy but does not converge to any real number L . #2 The goal of this problem is to ensure mastery of the basic concepts concerning limits of functions. Let where each of f , g , and h are defined on R , and k is defined on ....
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- Spring '08
- Topology, Continuous function, Metric space, Cauchy