8. Let f : (0,00) + (0,00), f(1) = V2 + . Let (an) be the sequence defined by ay = 1, 4n+1 = f(a), neN. Show that (an) c

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8 Let F 0 00 0 00 F 1 V2 Let An Be The Sequence Defined By Ay 1 4n 1 F A Nen Show That An C 1 (46.65 KiB) Viewed 62 times

8. Let f : (0,00) + (0,00), f(1) = V2 + . Let (an) be the sequence defined by ay = 1, 4n+1 = f(a), neN. Show that (an) converges and find the limit. Here are some steps you should include: (a) Show that ar> 1 for all n E N (Induction?) (b) Assuming the limit exists, determine what it would be. (c) Show that (an) is bounded above by (that is each term is less than or equal to the value you found in the previous step. (Induction?) (Hints: Consider a n+1---; what is the value in (b) squared?) (d) Show that (ar) is increasing. (Induction?) (e) Put everything together and reach a conclusion.