test4-fall94 - Name S.S.N MAS 3300 Test 4 Fall 1994 When...

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Name: S.S.N: MAS 3300 – Test 4 Fall 1994 When giving proofs quote the relevant axioms and results. This test has three parts: Part A Part B Part C Do at least ONE question from each part and do at least FIVE questions. The questions have equal value. If you do more than five questions please indicate which five you want graded as test questions. Any other question attempted will be marked as a bonus question. You may use your unmarked printed course notes. Calculators are allowed. Write on ONE side of your paper. 1
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PART A 1 . Prove that Z does not have an upper bound. [Hint: Use LUB] 2 . Prove that lim n →∞ 1 n 2 = 0; i.e. for each ε > 0 there is an integer N such that if n N then 0 < 1 n 2 < ε. 3 . Prove that between any two real numbers there is an irrational number. [You may assume the result that between any two real numbers there is a rational number.] 4 . Let a > 0. Define A := { x 0 : x 2 a } . It can be shown that A is bounded above and hence has a least upper bound u .
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