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MATH 104: INTRODUCTORY ANALYSIS SPRING 2008/09 PROBLEM SET 3 Unlike the previous problem set, in this one you will need to prove your claims rigorously. 1. (a) Prove Bernoulli’s inequality: (1 + x)n ≥ 1 + nx for every real number x ≥ −1 and every n ∈ N. (b) Define the sequence (an )n∈N and (bn )n∈N by 1 2n2 + 1 and b = . n 2n n2 + 3n Prove from the definition of limits that an = 2 − lim an = 2 = lim bn . n→∞ n→∞ In other words, given ε > 0, you should produce a corresponding N ∈ N that satisfies the definition. 2. Let (an )n∈N and (bn )n∈N be sequences of real numbers. (a) Suppose limn→∞ an = 0 and (bn )n∈N is bounded (but not necessarily convergent). Prove that limn→∞ an bn = 0. (b) Suppose (an )n∈N is convergent and (bn )n∈N is divergent. Prove that (an +bn )n∈N is divergent. 3. Let (an )n∈N and (bn )n∈N be sequences of real numbers. Are the following statements true or false? You need to prove the statement or give a counterexample. (a) If (an )n∈N is convergent and (bn )n∈N is divergent, then (an bn )n∈N is divergent. (b) If (an )n∈N and (bn )n∈N are both divergent, then (an + bn )n∈N is divergent. (c) If (an )n∈N and (bn )n∈N are both divergent, then (an bn )n∈N is divergent. 4. Let (an )n∈N be a sequence of real numbers. Define the sequence (sn )n∈N by a1 + a2 + · · · + an sn := n for every n ∈ N. (a) Prove that if limn→∞ an = a, then limn→∞ sn = a. (b) Give an example to show that the converse is not always true. Date: February 26, 2009 (Version 1.0); due: March 4, 2009. 1