314w08reviewproblems1 - 1 Write the definition of infimum...

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1. Write the definition of infimum by modeling it on the definition of supremum . State Lemma 1.3.7 for Infimums. 2. Let s = inf( A ) and define - A = {- x : x A } . Prove that sup( - A ) = - s . 3. Prove that inf ( { 1 + 1 n 2 : n N } ) = 1 using Lemma 1.3.7. 4. Find the limit of the sequence ( 2 , 1 + 2 , 1 + 1 + 2 , . . . ) [Hint: this sequence can be defined recursively.] 5. Find examples of the following, if possible: (a) a sequence which is Cauchy, but not monotone. (b) a sequence which is monotone, but not Cauchy. (c) a sequence which is bounded, but not Cauchy. (d) two sequences ( x n ) and ( y n ) such that x n diverges, y n converges, and ( x n + y n ) converges (e) two sequences ( x n ) and ( y n ) which both converge, but ( x n y n ) diverges. (f) two sequences ( x n ) and ( y n ) which both diverge, but ( x n y n ) con- verges. (g) A monotone sequence which diverges but has a convergent sub-
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