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

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

Expert Solution

Colby Messinger answered 2 years ago

If the sequence is increasing then it a) converges to its supremum b) diverges c) may...

If the sequence is increasing then it a) converges to its supremum b) diverges c) may converge to its supremum d) is bounded if S= { 1/n - 1/m: n,m belongs to N } where N is the set of natural numbers then infimum and supremum of S respectively are a) -1and 1 b) 0,1 c)0,0 d)can not be determined Please explain

Show that a monotone sequence converges if and only if it is bounded.

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].

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.

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.

Let f : [a, b] → R be a monotone function. Show that the discontinuity set...

Let f : [a, b] → R be a monotone function. Show that the discontinuity set Disc(f) is countable.

Show that the sequence an = (−1)^n doesn’t converge to 1 nor −1. Can it converge...

Show that the sequence an = (−1)^n doesn’t converge to 1 nor −1. Can it converge to anything other than 1 and −1?

Consider 100 g of 252Fm, which decays in a sequence of 5 α decays to eventually...

Consider 100 g of 252Fm, which decays in a sequence of 5 α decays to eventually reach 232Th. The relevant half-lives follow: 252 Fm :25.39 hr 248 Cf :333.5 d 244 Cm : 18.10 yr 240Pu : 6564yr 236 U :2.34 × 10^7 yr 232 Th :1.41 × 10^10 yr For the following questions, state clearly any approximations that you need to make. a) How much of the original sample is 252Fm and how much is 248Cf after one day?...

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.

1. What is a monotone class? 2.Prove that every algebra or field is a monotone class...

1. What is a monotone class? 2.Prove that every algebra or field is a monotone class Proof that 1. show that the intersection of any collection of algebra or field on sample space is a field 2. Union of field may not be a field

Question