Show that the sequence is Cauchy using the definition of Cauchy se- quences. Sn 2n +1...
Suppose that and are Cauchy sequences. Show that the sequence is also Cauchy. Sn We were unable to transcribe this image(Sn-tn
1. Prove that if {xn} is a sequence that satisfies 2n² + 3 Xnl73 +5n2 + 3 + 1 for all n e N, then {xn} is Cauchy. . Use the definition of limit for a sequence to show that 2. Suppose that {Xn} converges to 1 as n xn +1-e, as nº n
3. Give an example of a sequence {sn} that is not monotone, but the se- quence {s} is monotone. (7 points) carlo ST 4. Let $i = 4 and 9n+1 = (38m + 1)/5 for n 2 1. Show that the sequence {sn} is bounded and monotone, and find its limit s. (10 points)
(7) (14 pts). Use Cauchy sequence definition to prove {an) = {2:ne N} is a Cauchy sequence
Show that if an and bn are Cauchy sequences then anbn is a Cauchy sequence. Note that you are not allowed to use convergence, but you can use the definition and the fact that Cauchy sequences are bounded.
Show that, if an ≥ 0 for all n ∈ N and (an) is a Cauchy sequence, then (√ an) is also a Cauchy sequence. Hint: x − y = (√ x − √y)(√ x + √y) Show that, if an > 0 for all n є N and (an) is a Cauchy sequence, then (Van) is also a Cauchy sequence. Hint: r -y- (V1-vu) (Va + vⓙ Show that, if an > 0 for all n є N and...
4. (20 pts) Let {xn} be a Cauchy sequence. Show that a) (5 pts) {xn} is bounded. Hint: See Lecture 4 notes b) (5 pts) {Jxn} is a Cauchy sequence. Hint: Use the following inequality ||x| - |y|| < |x - y|, for all x, y E R. _ subsequence of {xn} and xn c) (5 pts) If {xnk} is a See Lecture 4 notes. as k - oo, then xn OO as n»oo. Hint: > d) (5 pts) If...
(1). Let {?n} be a Cauchy sequence in a metric spaces X and let {?n} be another sequence in X such that ? (?n, ?n) < 1/n ??? ????? ? ∈ ℕ. Show that {?n} is also a Cauchy sequence in X.
a. 1 4. Let Sn =EX=1 Show that In(n+1) < Sn S 1+In n b. Show that {an} = {Sn - In n} Show that sequence {an} converges C.
Suppose (Sn) is a sequence such that, for all n 1, we have 1 Sn+1 – snl < n2 Show that (sn) converges