Unformatted text preview: ** n s j =1 j 2 = n ( n + 1)(2 n + 1) 6 Note: This is book problem 3. 4. Suppose that S is a nonempty set of real numbers that is bounded. (a) Prove that inf S ≤ sup S . * (b) Why is it necessary that S be nonempty and bounded? * Note: Part (a) is book problem 13. 5. (a) Give an example of a set S of numbers that is nonempty and bounded above but has no * maximum. (b) Prove that a set S of numbers has a maximum i± it is bounded above and sup S ∈ S . ** Note: This is book problem 15 reorganized. 6. De²ne S = b x ∈ R v v x 2 < 2 x B . (a) Show that S is nonempty. * (b) Show that S has an upper bound. * (c) Which theorem can you apply and what is the conclusion? * (d) Conjecture a value for sup S . * (e) Show that sup S equals this value. ** Note: This is book problem 19 modifed and expanded....
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 Spring '08
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 Math, Mathematical Induction, J2, Book problem, sup S.

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