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Math 3515: Sample Midterm Exam Please complete any 6 of the 9 problems! More than 6 problems count as extra credit (specify your βbasic 6β). 1. Prove that if A is uncountable and B is countable, then A β B (A minus B) is uncountable. π 2. Prove that the set D of dyadic rationals, deο¬ned as π« = {π : π β π πππ π = 2π πππ π β π }, is countable. You may not use the fact that the rationals are countable. Note that the dyadic rationals are dense in R, i.e. between any two real numbers a and b there is a dyadic rational d such that a < d < b. This is just a nifty fact of life, nothing for you to do. 3πβ7 πββ π+2 3. Prove that lim =3 1 β 4. Find the inf, sup, lim inf, and lim sup for the sequence {(β1)π+1 + π } π=1 . Note: if ππ is the n-th prime number, then it is an unproven conjecture that πππ πππ ππ+1 β ππ = 2. Just another neat fact ο 5. Define a recursive sequence as follows: π1 = 0 and ππ+1 = β2 + ππ for n > 1. Show that the sequence must have a limit and find it. β β 2 2 6. Prove that if ββ π=1 |ππ | converges, then βπ=1|ππ | converges. Is the opposite also true, i.e. if βπ=1|ππ | converges then ββ π=1 |ππ | converges? 7. Are the following series absolute convergent, conditionally convergent, or divergent (justify your answer): a. ββ π=1 ( b. ββ π=1 π+1 π ) π (β1)π π 6+π2 1 π π c. ββ π=1(β1) ( ) 3 8. Find examples that illustrate each of the following questions: a. We know that any union of open sets is open. What about the union of closed sets, is that always closed? b. We know that if π΄π are compact and π΄π+1 β π΄π then the intersection ββ π=1 π΄π is not empty. What about the intersection of nested open sets, can that be empty? 9. Prove that if C is a compact subset of R, then sup(C) is part of the set. In other words if C is compact, then sup(C) = max(C).

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