m3a3_Prac4.pdf - Practice Problems 4 Not to be submitted...

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Practice Problems # 4Not to be submitted, but you are responsible for their content on Test 1, FridayFebruary 7, 1:30-2:20pm. Prepare these for tutorial the week of Feb 3.From the text:Sec. 3.5, p. 91, # 2, 5, 9Sec. 3.7, p. 100-101, # 3a, 4, 9b, 11, 12, 131A.For each sequence find all of the limit points (ie, the subsequential limits) andgive a convergent subsequence for each. What are lim supn!1xnand lim infn!1xn?(i)xn=(-1)nn1+n,8n2N;(ii)xn= 2[(-1)nn],8n2N;(iii)xn= sinn4,8n2N.B.Suppose the sequence (xn) converges tox(xn-!x) andyis a limit point (ie,the limit of a subsequence) of (yn). Show thatxyis a limit point of the sequence(xnyn)n2Nformed by multiplying the elements of the original two sequences.C. (i)Letxbe a (subsequential) limit point of (xn) andybe a subsequential limitpoint of (yn).Show (by example) thatx+ymight not be a limit point of the sequence (xn+yn)n2N.(ii)Find sequences (xn) and (yn) for which lim supn!1(xn+yn)6= lim supn!1xn+lim supn!1ynD.

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