Let (Xn) be a monotone sequence. Suppose that (Xn) has a Cauchy subsequence. Prove that (Xn)...

Let (Xn) be a monotone sequence. Suppose that (Xn) has a Cauchy subsequence. Prove that (Xn) converges.

Expert Solution

Colby Messinger answered 2 years ago

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 (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.

prove every cauchy sequence converges

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

Show that every sequence contains a monotone subsequence and explain how this furnished a new proof...

Show that every sequence contains a monotone subsequence and explain how this furnished a new proof of the Bolzano-Weierstrass Theorem.

Prove that if a sequence is bounded, then it must have a convergent subsequence.

Prove that every sequence in a discrete metric space converges and is a Cauchy sequence. This...

Prove that every sequence in a discrete metric space converges and is a Cauchy sequence. This is all that was given to me... so I am unsure how I am supposed to prove it....

4. Let a < b and f be monotone on [a, b]. Prove that f is...

4. Let a < b and f be monotone on [a, b]. Prove that f is Riemann integrable on [a, b].

(a) a sequence {an} that is not monotone (nor eventually monotone) but diverges to ∞ (b)...

(a) a sequence {an} that is not monotone (nor eventually monotone) but diverges to ∞ (b) a divergent sequence {an} such that {an/33} converges (c) two divergent sequences {an} and {bn} such that {an + bn} converges to 17 (d) two convergent sequences {an} and {bn} such that {an/bn} diverges (e) a sequence with no convergent subsequence (f) a Cauchy sequence with an unbounded subsequence

if (a_n) is a real cauchy sequence and b is also real, prove i) (|a_n|) is...

if (a_n) is a real cauchy sequence and b is also real, prove i) (|a_n|) is a cauchy sequence ii) (ba_n) is also a cauchy sequence

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