Math 3IA3 - Homework 2
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Math 3IA3 - Homework 2
Due: Wednesday, February 8, at 11:59 PM
Instructions (PLEASE READ CAREFULLY): For each question, please provide a full solution. If the question asks for an example or counter-example, please provide justification for your solution.
All solutions should be uploaded to Crowdmark with the correct orientation (i.e. please do not submit solutions that are sideways or upside-down). Please upload each solution separately in order to facilitate grading. Solutions type-set in LATEX are strongly encouraged.
Working in groups is not only allowed, it is encouraged, and completing the assignments on your own will probably be difficult. However, the final write-up should be completely in your own words. Please list the people you worked with on your final submission.
Due to the size of the course, not all solutions will be graded. The solutions to be graded will be chosen randomly, so it is in your best interest to do all of them. Each graded question will be graded out of 4 points, with points awarded for completeness, correctness, and readability.
1. Prove that 0 is the infimum of the set {1/n: n ∈ N, n ≥ 1}.
2. Prove that for every x,y ∈ R with x < y, there is some z ∈ R / Q such that x < z < y (in other words, the set of irrational numbers is dense in R).
3. Prove that the sequence (n2 ) does not converge. You can either do this from definitions or using theorems from lecture or the book.
4. Prove directly from the definition of convergence that limn →∞ (1/np ) = 0 for every fixed real number p > 0.
5. Prove that for any rational number x ∈ Q, there is a sequence (xn ) of irrational numbers, {xn : n ≥ 1} ⊂ R / Q, such that xn → x.
Hint: There are several ways to prove this using the previous problems .
6. Suppose that (sn ) is a sequence such that sn → 0. If (tn ) is a bounded sequence (not necessarily convergent), show that sn tn → 0.
7. Give an example to show that the boundedness assumption in the previous problem is neces- sary.
8. ⋆ Challenge problem – do not try to solve this alone at the last minute . Take your time and discuss with others. Prove that, if S is countable and S ⊆ T, then either T / S is finite or |T / S| = |T|.
Hint 1: In order to avoid making mistakes, consider the following examples . (a) T = N and S = {2k : k ∈ N}; (b) T = N and S = {k + 6: k ∈ N}; (c) T = R and S = N .
Hint 2: You may want to use the Cantor-Schr¨oder-Bernstein theorem, which is in the supple- mentary notes on Avenue .
Hint 3: This is different from the Hilbert Hotel problems, since T may be uncountable, but it could be helpful to think about the hotel.
2023-02-10