(ii) Theorem Suppose that {a} and {b} are sequences of positive numbers with bn = 0. Then limn→∞an = 0. an ≤ bn for all

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Ii Theorem Suppose That A And B Are Sequences Of Positive Numbers With Bn 0 Then Limn An 0 An Bn For All 1 (105.85 KiB) Viewed 12 times

(ii) Theorem Suppose that {a} and {b} are sequences of positive numbers with bn = 0. Then limn→∞an = 0. an ≤ bn for all N € N. Suppose that lim Proof: Let > 0 be given. Then there exists N N such that b₁ < € for all n ≥ N. So an < € for all n > N. Thus limnan = 0. a) Explain why (2) is true. b) Explain why (3) is true. c) Explain why (4) is true. 2 (1) and n ≤ yn for all n € N, then x ≤y. Thus liman <limn→ bn = 0. But each an >0 so limn→∞ an ≥ 0. Thus limnoo an = 0. (2) d) Suppose that a student gives the following argument in an attempt to prove the same result. Find the flaw in the reasoning and explain why the argument is not valid as stated. 004-11 (3) (4) If {n} and {n} are sequences of real numbers with lim x₁ = x and lim yn = y 11-00 (ii) (iv) [14 marks]