2. In this problem, we will explore a different notion of convergence. Consider the collection O of sets given by: 0 = {A CR: A° is finite} U {0} U {R} (a) Prove that a countable union of sets in O is also in O. (b) Prove that a finite intersection of sets in O is also in O. (c) Given a sequence (r,) of real numbers and a real number L, we will say that (x„) O-converges to L, if for every element O e O, we have that LE O implies that there exists an N € N such that r, E O for all n > N. Prove that the familiar sequence () O-converges to 0.

2. In this problem, we will explore a different notion of convergence. Consider the collection O of sets given by: 0 = {A CR: A° is finite} U {0} U {R} (a) Prove that a countable union of sets in O is also in O. (b) Prove that a finite intersection of sets in O is also in O. (c) Given a sequence (r,) of real numbers and a real number L, we will say that (x„) O-converges to L, if for every element O e O, we have that LE O implies that there exists an N € N such that r, E O for all n > N. Prove that the familiar sequence () O-converges to 0.

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

See similar textbooks

Similar questions

1) (i) Prove that every sequence converges to every point in a space with the trivial topology. (ii) Prove that no sequence converges to any point in a space with the discrete topology.
9. Given an infinite collection An, n = 1, 2, ... of intervals of the real line, their intersection is defined to be N=₁ A₂ = {x| (\n)(x = A₁)} Give an example of a family of intervals A₁, n = 1, 2, ..., such that An+1 C An for all n and 0. Prove that your example has the stated property. 1 An = Your answer needs to be a little bit longer. Write a few sentences to complete your assignment. n=1 10. Give an example of a family of intervals A₁, n = 1,2,..., such that An+1 C An for all n and 1 An consists of a single real number. Prove that your example has the stated property. Your answer needs to be a little bit longer. Write a few sentences to complete your assignment.
5
6. Which of the following statements is false in a metric space (X, d)? (a) Every Cauchy sequence is convergent. (b) Every open ball is an open set. (c) Every open set is a union of open balls. (d) Closed sets are complements of open sets. 7. Which of the following subsets of R is necessarily open under the discrete metric? (a) infinite sets (b) finite sets (c) intervals (d) all of the choices
5. Show that the following sequences of sets, {Ck }, are non-decreasing, (1.2.16), then find limä-∞ Ck- Ck = {x : 1/k < x < 3 – 1/k}, k = 1, 2, 3, ....
How should I show 2.31? Could you please include detailed explanation?
Show that the sequence (1/nk) nen is convergent if and only if k ≥ 0, nEN and that the limit is 0 for all k > 0.
Prove {(8n-5)/n} n=1 to infinity, is a Cauchy sequence in R with the usual metric dR(x,y) = abs(x-y)
Let's say we're in a proof and we know: (∀d)(S(d)→R(c,d))→(∀d)(M(d)→R(c,d)) Suppose we wanted to do universal instantiation, substituting scrat for the d's bounded by the first (leftmost) occurrence of (∀d). What is the result of this substitution? Group of answer choices (S(scrat)→R(c,scrat))→(∀d)(M(d)→R(c,d)) (S(scrat)→R(c,scrat))→(M(scrat)→R(c,scrat)) S(scrat)→R(c,scrat) (∀d)(S(d)→R(c,d)→(M(d)→R(c,d))) 2. In a proof, which of the following lines "push the stack?" (I.e., add to the number of vertical lines on the left-hand-side?) Group of answer choices Let n be an arbitrary natural number. Since a<b and a<b→0<c2, we can conclude 0<c2. Assume a<b. Since (∀c)R(c,c), R(e,e).
2. Let {an} be a bounded sequence in R and ƒ : R → R be a continuous and increasing function. Prove that lim sup f (an) = flim sup an n→∞ n→∞
(5) (from Royden) Show that the Monotone Convergence Theorem need not hold for monotonically decreasing sequences of nonnegative func- tions.