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Suppose a_n >0 for each n in N and lim inf (a_n) > 0. Prove there is a number a>0 st a_n >/= a for all n in N. (limit n--> infinity)
If {a_n + b_n} and {a_n} are both bounded, then {b_n} is bounded.
Consider the real sequence {x_n}_n generated by the iteration scheme x_n+1 = x_n(2-ax_n), for n = 0, 1, 2, ...... where a>0 and x_0 satisfying 0 < x_0 /=x_n>0 for all n. b. Prove x_n>/=x_n-1. c. Conclude that lim n-->infinity ...continues
Prove the following a) If lim n-->infinity (a_n*b_n) exists and lim n--> infinity (a_n) exists, then lim n -->infinity (b_n) exists. b) If lim n--> infinity (a_n) = 0 and {b_n} is bounded, then lim n-->infinity (a_n*b_n) exists and equals 0. c) If lim superior (a_n) exists, then {a_n}_n is bounded above.
Suppose that {a_n}_n is a real Cauchy sequence. Prove that lim superior n--> infinity (a_n) = lim inferior n--infinity (a_n) so as to conclude that lim n--infinity (a_n) exists.
Suppose the sequences {a_n}_n and {b_n}_n are both bounded above. a) Prove that for all n in the naturals sup{a_k + b_k: k>/=n} is less than or equals sup{a_k:k>/=n} + sup{b_k:k>/=n} b) Use this to conclude: limsup (a_n + b_n) is less than or equals limsup(a_n) + limsup(b_n) (all limits are n--> infinity)
Suppose that the sequences {a_n}_n is bounded above and lim(b_n) exists. a) Prove that for all e>0 there is an N st that for all n>=N sup{a_k:k>=n} + b_n <=sup{a_k + b_k: k>=n} + e. b) Use this to conclude limsup(a_n) + lim (b_n) <= limsup (a_n+b_n) (all limits are n ---> infinit ...continues
Suppose the summation from k=1 to n of a_k is absolutely convergent and {b_n} is bounded. Prove that this implies the summation from k=1 to n of a_k*b_k is absolutely convergent.
Consider the real number iteration scheme x_n+1 = f(x_n) for n = 1, 2, ... with x_1 given. In addition, suppose there is a number 0 < p < 1 st lf(x) - f(y)l < = plx-yl for all x,y. a) Show lx_n+1 - x_nl < = p^n-1lx_2 - x_1l for all n. b) From this, conclude {x_n}_n is Cauchy.
