Sets and Bounds # 1. For each set, fill in the table. State the infimum, minimum, maximum, and supremum (if they exist). If there is not a finite value or it does not exist, record an X instead. (To get a feel for the sets, you might want to rewrite them in different notation, or list some elements, etc.) Explanations not required. Set (a) Infimum Minimu m Maximu m Supremum {3 e , 1.2, 3 , /2,5 /4 } (b) {x R : 2 x < 8} (c) {x R : x 5} (d) {n N : |n + 1| } (e) {q Q : |q + 1| } (f) {x R : |x + 1| } (g) { (h) :n N } {5 n1 2n (1 )n 5 n+ n :n N 2 } (i) { ex : x R } (j) { arctan x : x R } (k) n ( { 3 ) :n N } (l) 1 1 n=1 4 , 9+ n n sin ( Page 1 of 11 (m) ( 1 1 n=1 4 , 9+ n n Page 2 of 11 #2. Let D be a nonempty set and suppose that f: D R and g: D R. Define the function f + g: D R by (f + g)(x) = f(x) + g(x). Recall that sets f(D), g(D), and (f + g)(D) are the ranges of f, g, and f + g, respectively. #2 (a) Fill in the blanks below to prove the following: If f(D) and g(D) are bounded above, then (f + g)(D) is bounded above and sup[(f + g)(D)] sup f(D) + sup g(D). Proof: Let D be a nonempty set, and suppose that f: D R and g: D R, and f(D) and g(D) are bounded above. Since D is nonempty, f(D), g(D), and (f + g)(D) are nonempty. Since f(D) and g(D) are nonempty and bounded above, by the ________________________(axiom/property), sup f(D) and sup g(D) must exist and be finite. Let u = sup f(D) and v = sup g(D). Then ______ u for all x in D and g(x) _____ for all x in D. Thus (f + g)(x) = f(x) + g(x) _________ for all x in ___, so (f + g)(D) is bounded above by __________. Since (f + g)(D) is nonempty and bounded above, by the ___________________________(axiom/property), sup[(f + g)(D)] must exist and be finite. So, the least upper bound for (f + g)(D) must be less than or equal to __________ (same as earlier blank) . Thus, sup[(f + g)(D)] sup f(D) + sup g(D). #2(b). Consider this statement: If f(D) and g(D) are bounded above, then sup[(f + g)(D)] = sup f(D) + sup g(D). Find a counterexample to show that this statement is false. [Thus you will be showing that back in part (a), " " could not be replaced by "=". ] Briefly explain your counterexample. Page 3 of 11 SEQUENCES REMINDER: A set is a collection of elements, with no order imposed. By contrast, a sequence is an ordered list. If "lim" is stated without an explicit direction, we will always assumed to be the limit of a sequence as n . Use the definition (page 2 of my Sequences document or page 44 of Lebl book) to carefully prove : #3. lim 9 cos ( 3 n ) =0. 4 n2 HINT: Is the cosine function bounded? Use the definition (page 2 of my Sequences document or page 44 of Lebl book) to carefully prove : #4. For any real number p > 0, (1)n lim =0 . np Page 4 of 11 Use the definition (page 2 of my Sequences document or page 44 of Lebl book) to carefully prove: lim #5. #6: Let 24 n+13 =4 6 n+7 1 s n= 1+ n n ( ) . It can be shown that (sn) is a bounded monotone sequence, so it converges. We designate the limit of this particular sequence by the number e. 1 n lim 1+ =e . That is, n n ( ) Use this result and limit theorem on page 6 of my Sequences document to find the limit of each of the following sequences. Show some work. (There should not be much work at all required!) (a) 1 n 6n ( ) s n= 1+ Page 5 of 11 (b) 1 n n ( ) s n= 1+ HINT: Rewrite the expression without a negative exponent. Page 6 of 11 #7. (a) Prove: If (an) is any sequence converging to 0, and (bn) is any bounded sequence, then lim (an bn) = 0. Hints: One approach is to show directly that the definition (page 2 of my Sequences document or page 44 of Lebl book) holds. Another approach is to show that the Theorem on page 4 of my Sequences document can be applied to get the desired result. Take your pick. #7(b). Disprove: If (an) is any sequence converging to 0, and (bn) is any sequence, then lim (an bn) = 0. Note that the difference between part (a) and part (b) is the omission of the word "bounded." Page 7 of 11 #8. For each of the following sequences (sn), state whether the sequence converges or diverges. If the sequence converges, determine the limit; otherwise leave the limit column blank. Show work or a brief explanation. You may cite textbook/(or my notes) theorems, examples, and/or homework/textbook exercises (by source and page number). No formal proofs required. However, it is not sufficient to merely list some terms of the sequence and state your conclusion without additional justification. Converge Or Diverge? Sequence Limit Work/Explanation n (a) (b) (1) s n= 4 n s n= 18 n3+ 5 2 3 n 20 n (c) (1)n n s n= 3 n7 (d) s n= n (1) n 5 n+3 Page 8 of 11 n 2 n1 (e) s n=1 (f) s n=4 n2 (g) s n= 8 n4 2 n2 +3 54n 45n Page 9 of 11 (h) (i) (j) s n=(n )n n s n=(5) s n= 7n n3 Page 10 of 11 (k) 7n s n= n! #9. Prove that the sequence (sn) given below is monotone and bounded. Then show how to find the limit algebraically. Let s 1=3 and 1 s n+1 =4+ s n 4 for n in N. HINT: See page 11 of Sequence notes and recursive sequence video http://sandsduchon.org/sands/MATH301/Videos/Chapter4/Chapter4Sec18Part2.swf. Page 11 of 11