Let {an}n∈N be a sequence with lim n→+∞ an = 0. Prove that there exists a...

Let {an}n∈N be a sequence with lim n→+∞ an = 0. Prove that there exists a subsequence {ank }k∈N so that X∞ k=1 |ank | ≤ 8

Solutions

Expert Solution

Solution:

Given That Let {an}n∈N be a sequence with lim n→+∞ an = 0. Prove that there exists a subsequence {ank }k∈N so that X∞ k=1 |ank | ≤ 8

Poof:

We have show that for each k we can choose an Ank so that |Ank| <1/2k.

Then the comparison test shows thatΣAnk is absolutely convergent, hence convergent.

Startwith k = 0. If there were no An0 with absolute value < 1,then 1 would be a lower bound for the whole of {|An|},contradicting the assumption that lim inf |An| = 0. Choose n0so that |An0| < 1.

Now there must be an n1 > n0 suchthat |An1| < 1/2, or else 1/2 would be a lower bound for{|An|}n > n0, and so lim inf |An| would be at least1/2, a contradiction.

Choose An1, and now, arguing the sameway, we can find n2 > n1 with |An2| < 1/4. Proceedingthis way, for every k we find Ank with |Ank| < 1/2k.

Colby Messinger answered 9 months ago

Next > < Previous

Related Solutions

Prove the following test: Let {xn} be a sequence and lim |Xn| ^1/n = L 1....

Prove the following test: Let {xn} be a sequence and lim |Xn| ^1/n = L 1. If L< 1 then {xn} is convergent to zero 2. If L> 1 then {xn} is divergent

Let {an} be a bounded sequence. In this question, you will prove that there exists a...

Let {an} be a bounded sequence. In this question, you will prove that there exists a convergent subsequence. Define a crest of the sequence to be a term am that is greater than all subsequent terms. That is, am > an for all n > m (a) Suppose {an} has infinitely many crests. Prove that the crests form a convergent subsequence. (b) Suppose {an} has only finitely many crests. Let an1 be a term with no subsequent crests. Construct a...

Let (xn) be a sequence with positive terms. (a) Prove the following: lim inf xn+1/ xn...

Let (xn) be a sequence with positive terms. (a) Prove the following: lim inf xn+1/ xn ≤ lim inf n√ xn ≤ lim sup n√xn ≤ lim sup xn+1/ xn . (b) Give example of (xn) where all above inequalities are strict. Hint; you may consider the following sequence xn = 2n if n even and xn = 1 if n odd.

Let F be a finite field. Prove that there exists an integer n≥1, such that n.1_F...

Let F be a finite field. Prove that there exists an integer n≥1, such that n.1_F = 0_F . Show further that the smallest positive integer with this property is a prime number.

Let (xn), (yn) be bounded sequences. a) Prove that lim inf xn + lim inf yn...

Let (xn), (yn) be bounded sequences. a) Prove that lim inf xn + lim inf yn ≤ lim inf(xn + yn) ≤ lim sup(xn + yn) ≤ lim sup xn + lim sup yn. Give example where all inequalities are strict. b)Let (zn) be the sequence defined recursively by z1 = z2 = 1, zn+2 = √ zn+1 + √ zn, n = 1, 2, . . . . Prove that (zn) is convergent and find its limit. Hint; argue...

Let {xn} be a real summable sequence with xn ≥ 0 eventually. Prove that √(Xn*Xn+1) is...

Let {xn} be a real summable sequence with xn ≥ 0 eventually. Prove that √(Xn*Xn+1) is summable.

let n belongs to N and let a, b belong to Z. prove that a is...

let n belongs to N and let a, b belong to Z. prove that a is congruent to b, mod n, if and only if a and b have the same remainder when divided by n.

Let x, y ∈ R. Prove the following: (a) 0 < 1 (b) For all n...

Let x, y ∈ R. Prove the following: (a) 0 < 1 (b) For all n ∈ N, if 0 < x < y, then x^n < y^n. (c) |x · y| = |x| · |y|

Let (an) be a real sequence in the standard metric. Prove that (an) is bounded if...

Let (an) be a real sequence in the standard metric. Prove that (an) is bounded if and only if every subsequence of (an) has a convergent subsequence.

Let a sequence {xn} from n=1 to infinity satisfy x_(n+2)=sqrt(x_(n+1) *xn) for n=1,2 ...... 1. Prove...

Let a sequence {xn} from n=1 to infinity satisfy x_(n+2)=sqrt(x_(n+1) *xn) for n=1,2 ...... 1. Prove that a<=xn<=b for all n>=1 2. Show |x_(n+1) - xn| <= sqrt(b)/(sqrt(a)+sqrt(b)) * |xn - x_(n-1)| for n=2,3,..... 3. Prove {xn} is a cauchy sequence and hence is convergent Please show full working for 1,2 and 3.

Subjects