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Question: number 1 please...

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Number 1 please
7796 (1 of 2) ou may work or discuss the questions with your classmates, but you must turn in your own rite-up, which should reflect your understanding. 1. Do the following sequences converge? If the sequence converges, find and prove what it converges to. If not, prove it does not converge (use the definition of convergence). (a) an= 21 +5 2. (a) Let (%) be a sequence, and suppose that r is a value that occurs infinitely often in the sequence (that is, the set S- nENn is infinite). i. Show that if sn converges, then the limit must be z. ii. Conclude that non-constant periodic sequences do not converge (non-constant meaning it takes on at least 2 values, and periodic meaning there exists k such that ai = ai+k for all i). (b) Fix θ E (, 1) n Q. Let as-sin (rt r) for n E N. i. Show that if θ-0, then an converges to 0. ii. Use part (a) to prove the following: if an converges, then θ 0 and an converges to 0 3. Iat (%) be a sequence satisfying s-, 0 for all n E N and s :limn-too 0. (a) First, prove that there exists N such that for all n > N 181 lsl (b) Use the result of part (a) to prove the following statement: if(%) is a convergent sexluence with snメ0 for all n E N, and limn→oos--s y, 0, then limi . (c) Use the result of part (b) to prove the following statement: if (sm) and () are two convergent sequences, where s 0 for all n E N, and lim Sa 0, then
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