UCLA MATH 131A - quiz2 (4 pages)

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quiz2



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quiz2

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Pages:
4
School:
University of California, Los Angeles
Course:
Math 131a - Analysis
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Math 131A Real Analysis Quiz 2 Instructions You have 50 minutes to complete the quiz There are three problems worth a total of 40 points You may not use any books or notes Partial credit will be given for progress toward correct proofs Write your solutions in the space below the questions If you need more space use the back of the page Do not forget to write your name and UID in the space below Name Student ID number Question Points Score 1 10 2 12 3 18 Total 40 Problem 1 10pts Suppose S is a nonempty bounded subset of R Then the set S defined by S s s S is also a nonempty bounded subset of R Prove that sup S inf S In fact they are equal but proving this is more work You can have 3 points for writing nothing On the other hand any severe logical fallacies which demonstrate writing down random nonsense in the hope of scavenging points will result in an immediate 0 Demonstrating that you understand relevant definitions what you need to verify to answer the question and clearly identifying any points at which you become stuck is a far better strategy Solution We will show that inf S is an upper bound for sup S Let s S By definition of S we have s S Since inf S is a lower bound for S we have s inf S Thus s inf S We have now demonstrated that inf S is an upper bound for S Since sup S is the least upper bound we have sup S inf S Problem 2 12pts Suppose that sn n 1 and tn n 1 are sequences of real numbers that 1 n N tn 2 and that sn and tn both converge to 1 Prove that the sequence sn tn n 1 converges to 1 For full credit you CANNOT use the algebra of limits without proof It is probably quicker to prove the result directly You can have 3 points for writing nothing On the other hand any severe logical fallacies which demonstrate writing down random nonsense in the hope of scavenging points will result in an immediate 0 Demonstrating that you understand how to write proofs which verify the relevant definition and clearly identifying any points at which you become stuck is a far better strategy Solution Let 0 Since sn converges to 1 there exists an N1 so that n N1 implies sn 1 4 Since tn converges to 1 there exists an N2 so that n N2 implies tn 1 4 Let N max N1 N2 and n N Then sn sn tn 2 sn tn 2 sn 1 1 tn 2 1 tn tn 4 4 Problem 3 Give counter examples or find problems with the following statements Answers should be short and to the point waffling will not be rewarded a 3pts If S 0 then sup S S b 3pts If S is a nonempty bounded subset of R then sup S max S c 4pts If S and T are nonempty bounded subsets of R and ST st s S t T then ST is bounded and sup ST max sup S sup T inf S inf T d 4pts Let T be a nonempty bounded subset of R and T 0 x R there exists an S T such that x sup S Then T 0 T sup T e 4pts If sn and tn are sequences of positive numbers and sn tn n 1 converges to 0 then sn tn n 1 converges to 0 Solution a sup S is the number 0 S is the set containing 0 b Some sets S do not have maximum elements S 0 1 is one such example c Let S 1 2 and T S 2 1 Then ST 4 2 1 So sup ST 1 But sup S sup T 2 1 2 and inf S inf T 1 2 2 d Let T 0 1 2 3 and T 0 x R there exists an S T such that x sup S Let S 0 1 Then S T and sup S 1 So 1 T 0 but 1 T sup T e Let sn n 1 and tn n Then sn tn n 1 converges to 1 and 1 6 0 1 sn tn Since n 1 n n 1 n 1 converges to 0 n


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