Exam 1
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Kuwait University Math 329 (Johnson) Dept of Math and Comp Sci Exam I
Date: March 21, 2001 Duration: 1 hr
Answer all of the following questions. Each (sub)question is worth 4 points. Calculators, pagers and mobile telephones are NOT allowed.
1. (a) Give an example of a nonempty bounded set S ⊂ R for which sup S 6∈ S. (b) Let T be a nonempty bounded subset of R with M = sup T . Prove that for every > 0, there exists t ∈ T such that M − ≤ t ≤ M .
2. Let F = {m2−n : m, n ∈ N}. (a) Prove that 0 is a limit point of F . (b) Prove that −1 is not a limit point of F .
3. (a) Prove that the sequence {cos(n2 − 5n + 3)} has a convergent subsequence. (b) Prove that if {cn } converges to C and {dn } converges to D, then {cn + dn } converges to C + D.
4. (a) Prove that (0, ∞) is an open set. (b) Prove that (−∞, 0] is a closed set.
5. Let the sequence {an } be defined recursively by a1 = 3, a2 = 4, 1 an+2 = an+1 + (an+1 − an ), 5 Prove that {an } is a convergent sequence.
n ∈ N.
Date: March 21, 2001 Duration: 1 hr
Answer all of the following questions. Each (sub)question is worth 4 points. Calculators, pagers and mobile telephones are NOT allowed.
1. (a) Give an example of a nonempty bounded set S ⊂ R for which sup S 6∈ S. (b) Let T be a nonempty bounded subset of R with M = sup T . Prove that for every > 0, there exists t ∈ T such that M − ≤ t ≤ M .
2. Let F = {m2−n : m, n ∈ N}. (a) Prove that 0 is a limit point of F . (b) Prove that −1 is not a limit point of F .
3. (a) Prove that the sequence {cos(n2 − 5n + 3)} has a convergent subsequence. (b) Prove that if {cn } converges to C and {dn } converges to D, then {cn + dn } converges to C + D.
4. (a) Prove that (0, ∞) is an open set. (b) Prove that (−∞, 0] is a closed set.
5. Let the sequence {an } be defined recursively by a1 = 3, a2 = 4, 1 an+2 = an+1 + (an+1 − an ), 5 Prove that {an } is a convergent sequence.
n ∈ N.
