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This final exam has 9 questions on 16 pages, for a total of 100 marks. Duration: 2 hours 30 minutes
Section Number (please circle): 201 / 202 Full Name (including all middle names):
Student-No:
Signature:
UBC Rules governing examinations:
- Each candidate should be prepared to produce his/her library/AMS card upon request.
- No candidate shall be permitted to enter the examination room after the expiration of onehalf hour, or to leave during the first half hour or the last 15 minutes of the examination. Candidates are not permitted to ask questions of the invigilators, except in cases ofsupposed errors or ambiguities in the examination questions.
- Candidates guilty of any of the following or similar practices shall be immediately dismissedfrom the examination, and shall be liable to disciplinary action: a) Making use of any books, papers or memoranda, other than those authorised by theexaminers. b) Speaking or communicating with other candidates. c) Purposely exposing written papers to the view of other candidates. The plea of accidentor forgetfulness will not be received.
- Smoking is not permitted during examinations. Question: 1 2 3 4 5 6 7 8 9 Total Points: 10 10 10 10 15 15 10 10 10 100 Score:
Please read the following points carefully before starting to write.
- Give complete arguments and explanations for all your calculations; answers withoutjustifications will not be marked — except where specifically stated.
- This is a closed-book examination.documents, cheat sheets or electronic devices of any kind (including calculators, None of the following are allowed: cell phones, etc.).
- You may not leave during the first 30 minutes or final 15 minutes of the exam.
- Read all the questions carefully before starting to work.
- Continue on the back of the previous page if you run out of space.
(f) Define the infimum of a set of real numbers
(g) Let f : A → B be a function and let D ⊆ B. Define the set f −^1 (D).
(h) Define what it means for the sequence {cn} to converge
(i) Define what it means for the sequence {cn} to diverge to infinity
(j) Let {cn} be a sequence. Define what it means for∑ n^ ∞=1 cn to converge
10 marks 2. (a) Letwhat values of α, β ∈ R and consider the function α, β is this function bijective? g : R → R defined by g(x) = α + βx. For
(b) LetProve that f : A → f B(C be an injective function and let C 1 , C 2 ⊆ A. 1 ∩^ C 2 ) =^ f^ (C 1 )^ ∩^ f^ (C 2 ). (c) Give an example to shows that the equation in (b) can fail when f is not injective.
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10 marks 4. Let f : (0, ∞) → R be a function defined by f (x) = x − (^1) x. (a) Prove that f is bijective. (b) Prove that |(0, ∞)| = |R|.
This page has been left blank for your workings and solutions.
15 marks 6. (a) Simplify as much as possible the nth^ partial sum of the series∑ k^ ∞=1 log
1 + k^1
(b) Prove, from first principles, that the sequence
{ (^) n (^2) + n − 3 n^2 + 4n + 5
converges to 1
10 marks 7. Let {an} be a convergent sequence with an → a. (a) Prove that there is some N ∈ N so that if n > N then |an| ≤ |a| + 1. (b) Hence (or otherwise) prove that that lim (a nn^ )^ = 0.
10 marks 8. Let {an} and {bn} be convergent sequences. Further, suppose an → a and bn → b. (a) Prove that if a 6 = b then there exists some N ∈ N so that if k > N then ak 6 = bk. (b) Now suppose thatthat a a = b. Prove or disprove that there is some k ∈ N so k =^ bk.
This page has been left blank for your workings and solutions.
