Question: a) Let a X. Prove that if xn = a for every n N, then xn converges. What does it converge to? b)

a) Let a ∈ X. Prove that if xn = a for every n ∈ N, then xn converges. What does it converge to?
b) Let X = R with the discrete metric. Prove that xn → a as n → ∞ if and only if xn = a for large

Step by Step Solution

★★★★★

3.48 Rating (165 Votes )

There are 3 Steps involved in it

1 Expert Approved Answer

Step: 1 Unlock

a If x n a for all n then px n a 0 is les... View full answer

Question Has Been Solved by an Expert!

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

Step: 2 Unlock

Step: 3 Unlock

Document Format (1 attachment)

741-M-N-A-D-I (557).docx

120 KBs Word File

Students Have Also Explored These Related Numerical Analysis Questions!

Suppose that xn > 0 and yn > 0 for all n N. Prove that if xn x as n (x may be an extended real number), then provided that none of these products is of the form 0 . lim sup(xnyn) x lim sup yn,...

Prove that if {xn} is a sequence that satisfies for all w N, then {x"} is Cauchy. 30 1.30 345n2 +3n + I

Let F(x) be the cdf of the continuous-type random variable X, and assume that F(x) = 0 for x 0 and 0 x + y | X > x) = P(X > y), Then F(x) = 1 ex, 0

4. (a) Let a E X. Prove that if Xn = a for every n E N, then Xn converges. What does it converge to? (b) Let X = R with the discrete metric. Prove that Xn ---t a as n ---t 00 if and only if Xn = a...

Section 10.5 Reading Assignment: Absolute Convergence; The Ratio and Root Tests Answer Only Exercise 1, 2, and 3 by using a screenshot provided Calculus Pearson textbook. Make sure you read these...

a.) Which of the following queues are restricted: i. ) (Xn), where Xn = n'tn for every n E N1. 2n2+1 ii. ) (Xn), where x1 * 0 and Xn+1 = -2xn for every n E N1. b.) Using the definition of a limit,...

Prove the second part of Theorem 3.8. That is let $\left\{X_{n}ight\}_{n=1}^{\infty}$ be a sequence of random variables, $X$ be a random variable, and $g$ be a Borel function on $\mathbb{R}$....

Suppose X; Y;X1;X2; : : : 2 L1. Prove the following statements. (a) Fatous lemma. If Xn 0 for every n and X D lim infn!1 Xn, then E.X/ lim infn!1 E.Xn/. Hint: By Problem 1.14, Yn WD infkn Xk is a...

Let n n 1 be a sequence of measures on a -algebra other words, suppose that for every A measure v on measure on , such that for every n , and let n converges to a function . In , lim n ( A) ( A) ....

Could someone please check my work against the definitions included Please state all definitions and theorems that you will need: Definition 3.4.2 Let a E R and let & > 0 . A deleted neighborhood of...

A labeled tree is one wherein the vertices are labeled. If the tree has n vertices, then {1, 2, 3,..., n} is used as the set of labels. We find that two trees that are isomorphic without labels may...

1. Vera Ltd manufactures table cloths. The company introduced a system of costing in January 2008 using the information of the previous year. The following is the budget for 2008: (i) Vera Ltd has a...

Kyrou Enterprises needs someone to supply it with 1 1 7 , 0 0 0 cartons of machine screws per year to support its manufacturing needs over the next five years, and you ve decided to bid on the...

(Absolute and Comparative Advantage)You have the following information concerning the production of wheat and cloth in the United States and the United Kingdom: Labor Hours Required to Produce One...

9. First prove that the radius of convergence of I:~o akxk is ~ 1.

8. Use the method of Example 7.36 to estimate 1f'(x)l.

4. Use Exercise 8 in Section 2.5 to prove that if lim sup lak/ak+ll

EBIT Sensitivity Stewart Industries sells its finished product for $8.71 per unit. Its fixed operating costs are $19,700, and the variable operating cost per unit is 54 29. a. Calculate the firm's...

The table below shows historical annual returns for stocks of companies X and Y over recent years. Calculate the arithmetic average returns. In addition, calculate their historical variances, as well...

Monsters Incorporated (MI) is ready to launch a new product. Depending upon the success of this product, MI will have a value of either $100 million, $150 million, or $191 million, with each outcome...