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hw1 - n s j =1 j 2 = n n 1(2 n 1 6 Note This is book...

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MATH 410: Homework 1 Due in class Friday 2/3/2012 Section 1.1 1. For each of the following statements determine whether it is true or false and justify (infor- mally) your answer. (a) The sum of irrational numbers is irrational. * (b) The product of irrational numbers is irrational. * (c) If n is natural and n 2 is odd then n is odd. * Note: These are book problems 1c,d,e. Note: Generic statements like these are for-all statements. 2. For each of the following statements determine whether it is true or false and justify (infor- mally) your answer. (a) Every nonempty set of real numbers that is bounded above has a largest member. * (b) If S is a nonempty set of positive real numbers then 0 inf S . * (c) If S is a set of real numbers that is bounded above and B is a nonempty subset of S then * sup B sup S . Note: These are book problems 2a,b,c. 3. Use the Principle of Mathematical Induction to prove that
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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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