Question: Prove that if xn,yn are bounded sequences with positive values, and yn converges, then lim sup(xnyn) = (lim supxn)(limyn). Prove that if (xn)n and (yn)

Prove that

if xn,yn are bounded sequences with positive values, and yn converges, then

lim sup(xnyn) = (lim supxn)(limyn).

Prove that

if (xn)n and (yn) are such that{ nN: xn different to yn} is a finite set, then the series sum from n=1 to infinity xn and sum from n=1 to infinity yn either both converge or both diverge. (Hint: use the Cauchy criterion.)

Step by Step Solution

There are 3 Steps involved in it

1 Expert Approved Answer

Step: 1 Unlock blur-text-image

Question Has Been Solved by an Expert!

Get step-by-step solutions from verified subject matter experts

Step: 2 Unlock

Step: 3 Unlock

Students Have Also Explored These Related Mathematics Questions!

Let xn and yn be bounded sequences. Suppose that the sequence xn is convergent and let x = lim (n to infinity) xn. prove that: lim (n to infinity) sup (xn + yn) = x + lim sup (n to infinity) yn

Show work/explanation where indicated. Answers without any work may earn little, if any, credit. You may type or write your work in your copy of the exam, or if you prefer, create a document...

1. (60 pts True/False. Indicate whether the statement is true or false. If the statement is false, present an explanation or counterexample. (No explanation necessary for each true statement.) (2...

1 For each of the two sequences, determine if it converges. If it does, provide the limit and prove the convergence. If it doesn't, provide a proof that the sequence diverges. a. xn = n ^2/5 /(n^...

A) Let {xn } be a bounded sequence. Prove directly using the denition of lim inf and lim sup that there exists two possibly dierent subsequences {xnk } and {xnk } of {xn } where lim inf xk = lim xnk...

The Basics of Financial Mathematics Spring 2003 Richard F. Bass Department of Mathematics University of Connecticut c These notes are 2003 by Richard Bass. They may be used for personal use or class...

MA250 - Homework Assignment #3 (Fall 2016) Due: Friday Oct. 7, 2016 (at the beginning of class) Instructor: Dr. Chester Weatherby Instructions: Complete the following problems and hand in a neatly...

Let $\left\{X_{n}ight\}_{n=1}^{\infty}$ and $\left\{Y_{n}ight\}_{n=1}^{\infty}$ be sequences of random variables and let $\left\{y_{n}ight\}_{n=1}^{\infty}$ be a sequence of real numbers. a....

4. Let X1, X2, . . . denote a sequence of random vectors with ||X,|| 0, there exists a constant Me > 0 for which P(IXAll > ME) 1 is bounded in probability. Prove the following facts for Xn, Yn E Rd....

Let 0

By referring to Figure 13.18, determine the mass of each of the following salts required to form a saturated solution in 250 g of water at 30 oC: (a) KClO3, (b) Pb(NO3)2, (c) Ce2(SO4)3.

Explain the difference in assumptions in the Gordon's growth model and Solomon's model in the calculation of share price (2mks) b) Kirengo Company Ltd is expected to announce a dividend per share of...

What is the primary difference between the Business Income ( Without Extra Expense ) Coverage Form and the Business Income ( And Extra Expense ) Coverage Form? Group of answer choices There is no...

Pemowe Conditions formats C will (11) wners of Tellmart Shoe Store are contemplating selling the business. The cumulative earnings for the past five years amounted to $750,000, including a on...