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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 β€œbasic 6”).
1. Prove that if A is uncountable and B is countable, then A – B (A minus B) is uncountable.
2. Prove that the set D of dyadic rationals, defined as 𝑫 = {π‘ž : 𝑝 ∈ 𝑍 π‘Žπ‘›π‘‘ π‘ž = 2𝑛 π‘“π‘œπ‘Ÿ 𝑛 ∈ 𝑁 }, 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.
π‘›β†’βˆž 𝑛+2
3. Prove that lim
1 ∞
4. Find the inf, sup, lim inf, and lim sup for the sequence {(βˆ’1)𝑛+1 + 𝑛 }
Note: if 𝑝𝑛 is the n-th prime number, then it is an unproven conjecture that π‘™π‘–π‘š 𝑖𝑛𝑓 𝑝𝑛+1 βˆ’ 𝑝𝑛 = 2. Just
another neat fact 
5. Define a recursive sequence as follows: 𝑏1 = 0 and 𝑏𝑛+1 = √2 + 𝑏𝑛 for n > 1. Show that the sequence must
have a limit and find it.
6. Prove that if βˆ‘βˆž
𝑛=1 |π‘Žπ‘› | converges, then βˆ‘π‘›=1|π‘Žπ‘› | converges. Is the opposite also true, i.e. if βˆ‘π‘›=1|π‘Žπ‘› |
converges then βˆ‘βˆž
𝑛=1 |π‘Žπ‘› | converges?
7. Are the following series absolute convergent, conditionally convergent, or divergent (justify your answer):
a. βˆ‘βˆž
𝑛=1 (
𝑛+1 𝑛
(βˆ’1)𝑛 𝑛
1 𝑛
c. βˆ‘βˆž
𝑛=1(βˆ’1) ( )
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 𝐴𝑛 are compact and 𝐴𝑛+1 βŠ‚ 𝐴𝑛 then the intersection β‹‚βˆž
𝑛=1 𝐴𝑛 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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