MATH 210 Exam 2, Fall 2017

MATH 210: Introduction to Analysis Fall 2015-2016, Midterm 2, Duration: 60 min. Exercise 1. (a) (10 points) State the definition of a metric space. (b) (10 points) Prove that (R2, d∞) is a metric space. Exercise 2. (1) (8 points) Prove that the series � n≥1 an with an = 1 √n + 1 − 1 √n is convergent and compute its sum. (2) (12 points) Show that the series � an n converges if and only if −1 ≤ a < 1. (3) 5 points. Assume that the series � a2 n and � b2 n converge. Prove that the series � anbn converges absolutely. Exercise 3. (a) (10 points) Find the interior and the closure of Z. Explain clearly to ensure full credits. (b) (10 points) Show that {(x, y) ∈ R2 | x2 + y2 ≤ 1} ∩ {(x, y) ∈ R2 | y ≥ x2} is compact. Explain clearly to ensure full credits. Exercise 4. We say that a sequence {(an, bn)} of R2 blows up if d∞((an, bn), (0, 0)) diverges to +∞. (a) (4 points) Prove or disprove using an explicit counterexample that if a sequence {(an, bn)} blows up then |an| and |bn| diverge to +∞. (b) (4 points) Prove that if a subset A ⊂ R2 is unbounded then there is a sequence {(an, bn)} of elements of A that blows up. (c) (4 points) Prove that if a subset A ⊂ R2 contains a sequence {(an, bn)} that blows up then A is unbounded. (d) Prove that the following sets are unbounded. i. (4 points) R × (0, 1) = {(x, y) ∈ R2 | 0 < y < 1}. ii. (4 points) {(x, y) ∈ R2 | x2 = y2}. Exercise 5. Let {xn} be a sequence of real numbers. (a) In this question, we suppose that the series � |xn+1 − xn| converges. The goal of this question is to prove that {xn} converges. Denote by SN the partial sum of � |xn+1 − xn| of order N. Let ε > 0. i. (5 points) Explain very briefly why there an integer N0 such that for any integers M, N ≥ N0 we have |SM − SN| < ε. ii. (5 points) Assume that M ≥ N. Show that |SM − SN| = |xN+1 − xN| + |xN+2 − xN+1| + · · · + |xM+1 − xM|. 2 iii. (5 points) Deduce that for M ≥ N ≥ N0 we have |xM+1 − xN| < ε. (hint: write xM+1 − xN as a telescopic sum). iv. (5 points) Deduce that {xn} converges. (b) (5 points) Give an example of a sequence {xn} such that |xn+1 − xn| converges to zero but {xn} diverges.