12.4 Show lim sup(sn +tn) < lim sup s, +lim sup tn for bounded sequences (Sn) and (tn). Hint: First show sup{sn + tn : n > N} < sup{sn :n > N}+sup{tn :n > N}.

Q: Let (n) be a bounded sequence. (a) Prove that lim sup(x) = limm- (b) Prove that lim inf(x) = limm-…

A:

Q: 12. Suppose (an) is a Cauchy sequence, and that (bn) is a sequence satisfying lim,co lan - bn| = 0.…

A:

Q: Let (sn) be a sequence with lim sup sn = 10 and lim inf sn = 5. (a) Show that (sn) is bounded. (b)…

A: Consider a sequence, Sn, with lim sup Sn = 10 and lim inf Sn = 5. (a) To show that Sn is bounded. By…

Q: 16. Show that if (s} and (ta} Cauchy sequence. Cauchy sequences, then {s + tn} is also are a.

A: Consider the provided question, We have to show that if {sn} and {tn} are Cauchy sequence then {sn +…

Q: how that every Cauchy sequence {pn} is bounded.

A:

Q: Let (an) be an unbounded sequence. Show that it has a subsequence (bk) = (ank) 1 such that lim = =…

A: Given, an  unbounded sequence.It is required to prove there exists a subsequence  such that .To…

Q: 1 a) Sn+1 = 3 – Sn 25, -35m b) Sn+1 Sn-1 9Sn c) Sn+1 S+1

A:

Q: 2 then E " Cn+1 converges. 3.11 limnr+m

A:

Q: Show that if (xn) and (yn) are bounded sequences, then lim sup(x, + Yn) < lim sup(x,) + lim sup(y,)

A:

Q: 7. Use the ratio test for sequences to prove that limn→∞ n¹/n² = : 1.

A:

Q: Exercise 8. Let {an}-1 be a sequence such that an lim L+0. n-00 n Prove that {an}1 is not bounded.

A: Given that &lt;an&gt;is a sequence such that  limn→∞ann=L≠0 To prove: &lt;an&gt; is not bounded.

Q: Let (an) and (bn) be two bounded sequences of positive real numbers. Prove that lim sup(anbn) (lim…

A: Given two bounded sequences of positive real numbers  and .We have to prove that, .To do this, use…

Q: 7

A:

Q: Problem, A circle whose area is 100t m2 is cut into four segments by a vertical and horizontal chord…

A:

Q: w/ バ+n 3n't -5- an- anD 70+4n? 3n-1

A:

Q: 5 Let (sn) be a bounded sequence, and let k be a non number. onnegative real (a) Prove lim sup(ksn)…

A: Let Sn be a bounded sequence and let K be a non-negative real no. (a)  Let limn→∞Sup Sn=l Let K&gt;0…

Q: A sequence ₁,₂... is such that u, = -r-1 (i) (ii) n Show that Σ r=2 11+5 Hence find Σ r=8 =2_n+1 n!…

A:

Q: o prove that knVA=| Test the seuence (ConViordív) of (10) an = (^_-) ) an 15 ら 9n: 3ntl 3n-1 (の+ (7)…

A:

Q: can you show your work on the paper so i can understand

A: To prove:liminfn→∞​yn​⋅(limsupn→∞​yn​xn​​)≤limsupn→∞​xn​ Given that  (xn) and (yn) are two positive…

Q: 32. Find the limit of the series: x (1+x) (+x) 1+x2 A) B) xx X+x C) (1+x D) (1+x) E) x

A: Given series is,

Q: 3.5.4) Show directly from the definition that of (xn) and (yn) Cauchy sequences, then cxntyn) and…

A:

Q: A momotone sequence converges if and only if it is bounded ylgn İhi

A: True

Q: +oo and inf{tn :n e N} > -∞, 11 (a) Show that if lim sn lim(sn + tn) = +o0. then (b) Show that if…

A: This question related to sequence from real analysis.

Q: without using the Cauchy Completeness Theorem show that if (sn) and (tn) are both Cauchy sequences,…

A:

Q: 2) The purpose of this problem is to give you some practice with the most common application of The…

A:

Q: 5. Using the limit method, show that every polynomial p(n) = a¿n² + ak- belongs to O(n*). ... + ao,…

A: Big theta notation is an order notation. It determines the growth rate of two growth functions. It…

Q: The producat of two divergent sequences is divergent. Select one: OTrue OFalse

A: False

Q: 6. (a) A sequence {anneN is said to be bounded if there exists M > 0 such that an M for any n E N…

A:

Q: Q3/ Find out the convolution of two sequences given below: X(n) = x1(n) * x2(n) X1(n) = { 1, 0 , -3…

A:

Q: Let (an) and (bn)1 be two sequences. Assume that the sequence (Σ1 ak)=1 is bounded, and that (bn) is…

A:

Q: Show that lim inf a ≤ lim sup a, for any sequence in R n

A: Suppose that {an} is unbounded, then either lim sup an=∞ or lim inf an=-∞. Hence, in case of an…