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

Q: Give an example of a bounded sequence {a„}_1 having exactly 2 limitpoints.

A: we have to find a sequence such that it has two limit points.

Q: Let x, = 3. Using the algebra of limits and standard results on limits to establish/justify if the…

A: Consider a sequence an. Then if the limit limn→∞an exist then the sequence converges otherwise the…

Q: Let {xn} be a bounded sequence on R. Show that lim = 0. n→+∞ n

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Q: Let (h,), (tn) be sequences of bounded functions on A that converges uniformly on A to h, t…

A: Given that there are two sequence (hn) and (tn) be sequence of bounded function on A that converges…

Q: 5n 9. Use the L'Hopital's rule to determine if the sequence an = () converges or diverges as limit n…

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

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Q: . Suppose real sequences (S,) and (t,) are bounded. (That is, that their ranges are bounded sets.)…

A: Given the real sequences sn and tn are bounded. We have to show that The sequence given by sn + tn…

Q: + R be defined by fn(x) = an Find the pointwise limit in 1.3 converges uniformly on [–3, 3]?…

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Q: Let {xn} be a bounded sequence on R. Show that Xn lim = 0. - n→+o n

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Q: Determine the limit of the sequence a, u UIS 3 Ve Divergent 4 1 5

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Q: Evaluate the limit using techniques from Chapters 2 and 4 and using L'Hoptal's Rule. 3(x-7) Iim x…

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Q: In (x) — х + 1 lim x² 2х + 1 x→1

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Q: 3] Let (xn) be bounded (not necessarily convergent) sequence, and assume that lim(yn) = 0. Show that…

A: A sequence is said to be bounded if it is bounded above and below by a number. That is, the sequence…

Q: Let (an). (bn) be two bounded sequences. a. Show that lim sup max(a,, b,) > max(lim sup an, lim sup…

A: Given: Let's say there are two bounded sequences (an, bn) .a. Demonstrate that limn→∞sup max(an,…

Q: Define the sequence {Sn} by Sn= [ 1 / (2)(1) ] + [ 1 / (3)(2) ] +   .............  + [ 1 / (n+1)(n)…

A: The nth term of the partial sum is given by

Q: 6. Let (x,) be properly divergent and let (y,) be such that lim(x,y,) belongs to R. Show that (y,)…

A: To show that (yn) converges to 0, given (xn) is properly divergent and (yn) is such that lim(xnyn)…

Q: Find the Z-transform of the following sequence/function (with ROC). Show complete solution.

A: Z-transformation is a discrete transformation of signal. Through the Z-transformation, one can…

Q: Give an example of a bounded sequence that has a limit.

A: Give an example of a bounded sequence that has a limit.

Q: Evaluate the limit using L'Hopital's Rule. In z lim IH00 In (z+1)* 4 1 00

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Q: Let (xn) be a bounded sequence. Recall that lim inf r, = sup inf xm = sup inf{xn, Xn+1,· ·}, lim sup…

A: Part(a): Let, L=limn→∞ inf xn=x then x is finite since xn is finite since xn is bounded. Now, for…

Q: Show directly from definitions that if sn and tn are sequences so that limn→∞ sn = 3 and limn→∞ tn =…

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Q: 4n+1 Σ Z 3n – 2 3п — X=1

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Q: Suppose that (sn) and (tn) are sequences so that sn = tn except for finitely many values of n. Using…

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Q: Evaluate the limit using Taylor series.

A: Given : The limit is limx→0+ cos x-12x .

Q: 5xn X1 = 1; xn+1 = 1; Χn+1

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Q: Obtain the DFT of the given sequences using Direct method x(n) = {4,1,1,2}

A: Introduction: The discrete Fourier transformation is used to transform a discrete signal. Discrete…

Q: Let (hn),(tn) be sequences of bounded functions on A that converges uniformly on A to h,t…

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Q: Show that the sequence of functions on the domain [-1, 1] defined by if -1<r<0, 0, if 0 <x < 1/n, 1,…

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Q: Prove or disprove that if a sequence {x,} satisfy x,+1 – xn → 0 then it is Cauchy

A: We have to solve given problem:

Q: Show directly from the definition that if (x„) and (yn) are Cauchy sequences, then (X, + yn) and…

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Q: Give an example of two bounded sequences {a„}1; {bn}%=1 such that lim sup„→»(@n + bn) # lim sup,„→∞…

A: Consider the sequences: an=(-1)n and bn=(-1)n+1 Then clearly, an≤1 and bn≤1Therefore, both the…

Q: Evaluate the limit using L'Hopital's Rule. 3 3 lim e -1 00 3 3 2

A: Please refer the attached image for complete solution.

Q: 4. Suppose that lim s, =0 and suppose that (t, ) is a bounded sequence. Show that lim s,t, = 0 %3D

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Q: prove that if X =CXn) is bounded increasing sequence of real ConVergent and Lim (xu) - Sup [An ineN]…

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Q: If {xn} is a bounded sequence on R. Prove: lim n→+∞

A: Given  sequence xn is bounded  we want  to show that  limn→∞xnn=0

Q: Use Theorem 1 to determine the limit of the sequence or type DIV if the sequence diverges. an 49n2…

A: Givenan=2n49n2+5

Q: Show that if f is continuous but not uniformly continuous on (a,b), then there is a Caushy sequence…

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Q: Vx+14-4 Compute the limit analytically without L'hopital's rule: lim X-2

A: Given limit is : limx→∞x+14-4x-2 To find the value of limit.

Q: Exercise 2. Use the definition of limit to prove that both of the sequences {+} and {"} converges to…

A: NOTE: We’ll answer the first question since the exact one wasn’t specified. Please submit a new…

Q: Show that the limit of a convergent sequence in a metric space is unique

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Q: 1. Let f: (0, 1) R be uniformly continuous. Let (r) be a sequence in (0, 1). Suppose that lim, x In-…

A: We will show lim f(xn) exists. For this we will show f(xn) is a Cauchy sequence of real numbers.

Q: 8. Prove that lim(x_) = 0 if and only if lim(jx,i) = 0. Give an example to show that the conver-…

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Q: The sequence (Xn) = (1, 3, 1, 3, 1, . . . , 1, 3, ...) is: Converge to 3. O Divergent. Converge to 0…

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Q: bn = /n In ( 1 + %3D

A: NOTE: Refresh your page if you can't see any equations. . the given sequence is  find the limit

Q: If a sequence c1, C2, C3, . . . has limit K then the sequence e1, e2, e3, . .. has limit e“. Use…

A: Answer

Q: . Let ƒ : (0, 1) → R be uniformly continuous. Let (Tn) be a sequence in (0, 1). Suppose that lim, s0…

A: Given that   f:(0,1) →ℝ be uniformly continuous function. And (xn)∞n=1 be a series in (0,1) such…

Q: Use the definition of the limit of a sequence to prove that lim (Vn + 2021 – vn) = 0. that jat- %3D

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Q: Show that the sequence {xn}: cost -dt is a Cauchy sequence. t2

A: The given sequence is xn=∫1ncostt2dt. Cauchy sequence:  A sequence xn of real numbers is said to be…