2. Let an be the sequence defined inductively by a₁ = 2 and an+1 = 2 2 an + an (a) Prove by induction that an € [1, 2] for all n € N. (b) Prove that a ≥ 2 for all n € N. (c) Hence prove that the sequence is decreasing. (d) We already know that (an) is bounded below (by (a)) so it follows, by the mono- tone convergence theorem, that (an) converges to some limit L. Show that L> 0 and L² = 2.

Q: 18. Which of these sequences is/are not bounded? I. a, = (-1)", n> 1 А. II only В. I only С. П only.…

A: Option (C) is correct. II only.

Q: Let x₁ = √2, and x1 Xn+1 = 2 + √√xn, n = 1, 2, 3, .... Prove that the sequence {n} converges, and…

A: Let  and To Find:Prove that the sequence is monotonically increasing and bounded above also shows…

Q: 21

A:

Q: 16.) In 2₁ [x], compute ftg (Note: each sequence should only be 0, 1, 2, 3, 4, 5, 6, 7). f(x) =…

A: (16) Given that , fx=x2-3x+2 and gx=2x3-4x+1 find f+g in Z7x.

Q: a) Suppose (pn) has limit L, and let S be the set of values of the terms of (pn). Can we say that L…

A:

Q: Prove that if {xn} is a sequence such that 1+ n |æn| < for all n E N, 1+ n2 then the sequence {xn}…

A:

Q: 1 2 + xn (a) Prove by induction that în > 0 for all n € N. (b) Prove that (n) is a contractive…

A: (a) We have to prove by induction that xn &gt; 0 for all nЄ N.(b) We have to prove that (xn) is a…

Q: Prove that x ∈ E' iff there exists a sequence (xn) in E\{x} such that xn -> x.

A:

Q: (n + 1)! Determine the limit of the sequence a, n! 3 1 2 1 1

A: find limit of sequence

Q: 14. Prove that every Cauchy sequence is bounded (Theorem 1.4)

A: To prove: Every Cauchy sequence is bounded.

Q: Choose the correct answer. 1. The sequence (x) = (-n) is. a) Cauchy b) bounded e) divergent d)…

A: We have1) The sequence (xn)=(-n) is . . . . . . . . .Now we have take n=1 x1=-1put n=2x2=(-2) So, we…

Q: Consider the sequence (an) that is defined inductively as a1 = 1 and an+1 = an(1+1/n4 ) i) prove…

A:

Q: 40. Which of the following functions can be used to find the nth term of the sequence 6, 11, 16, 21,…

A:

Q: Prove that the sequence {f(1/n)} has a limit if |f'(x)| < 1 for all r € (0, 1]. I

A:

Q: 1. Give an example of sequence of real numbers with the following properties.  a) Cauchy but not…

A:

Q: 2. Let (2n) and (yn) be Cauchy sequences of real numbers. Define (Tn) to be equivalent to (yn),…

A: Given :  xn and yn be the Cauchy sequences of real numbers. And  (xn) ~ (yn)  if  limn→∞xn-yn = 0 To…

Q: A and the set B = {x, y}, then A X B is denumerable as well.

A:

Q: For a sequence (an), its lower and upper limits are defined by lim inf an = sup inf am = sup inf{an,…

A:

Q: Suppose {f,} is a sequence in Banach space B s.t. ||fn - fn+1|| s'/2-). Show that the sequence {f,}…

A: For a vector space X, over a field F, a normed linear space X,· is referred to as Banach space if…

Q: == 2.9 Let (an) be a sequence with limit a, and define bɲ = an+1 (n = 1, 2,...). Show that (bn) → a.

A: We know that  A sequence an converges to a limit L  if, for any positive real number epsilon there…

Q: 3. Prove that if (a, } and (b,} are Cauchy sequences of rational numbers satisfying lim (a, - b, = 0…

A: 3.

Q: 2. Let an be the sequence defined inductively by ai = 2 and an+1 = 1 an + 2 an (a) Prove by…

A: Disclaimer: Since you have posted a question with multiple sub-parts, we will solve first three sub…

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

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Q: Prove {an} is decreasing for n >>1, if ao=1 an

A: A sequence {an} is said to be decreasing when we have  an+1-an&lt;0

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: 4. If {Cn} is a convergent sequence of real numbers, does there necessarily exist R > 0 such that…

A: Given that cn is a convergent sequence of real numbers and let it converge to a limit l. Then by…

Q: 2. Let (an)1 be a sequence of integers. Prove that (an) converges if and only if there exists a…

A:

Q: 2. Prove the sequence an = n"/n!, n > 1, is %3D (a) increasing; (b) not bounded above (show an 2 n).

A:

Q: (2) Let an be the sequence defined by: a₁ = 1 a2 = 2 an an-1 + 2an-2 for n ≥ 3. (a) Find a3,.. (b)…

A: Given that a1=1 , a2=2 and an=an-1+2an-2  for n≥3

Q: Prove that az E E' iff there exists a sequence (en) in E\{x} such that xnx.

A:

Q: Suppose SELECT ALL A) sometimes converges depending on the sequence, a(n) B) lim a(n) # 0 never…

A: Given that, limn→∞a(n)≠0 then we have to find ∑n=1∞a(n) is convergence or divergence or depends.