Describe the Dedekind cut corresponding to 2. Show that 2 + 2 = 4 on the level of Dedekind cuts. Explain why the intersection in a metric space M of a closed subset with a sequentially compact subset is sequentially compact. Give an example of a sequence an so that lim sup an does not agree with limn→∞ an, or explain why no such sequence exists. Give an example of an increasing function f : R → R (everywhere de- fined) that fails to be continuous at at least 3 points.

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A: Let  be recursively defined by ,  for .Find .

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A: We will find the limit.

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A: Step 1: Let s=supS.  So,  for any ε>0, there exists an x∈S such that s−ε<x.  Therefore, for…

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Q: 38

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Q: Use the La`Hospital rule to evaluate

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Q: (a) g: Zg → Z6 defined by g(n) = n 2. (b) g: Z6 → Z6 defined by g(x) =x €x. (c) g: Zs → Zs defined…

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