Description

4 attachmentsSlide 1 of 4attachment_1attachment_1attachment_2attachment_2attachment_3attachment_3attachment_4attachment_4

Unformatted Attachment Preview

Let

X:=

=

{(an)n21 CR; ƏN E N such that yn>N:an = 0},

and let doc((an)n21, (bn)n21) := supn=1 lan – bnl.

– .

in

(a) (3 points) Prove that (X, doo) is a metric space.

(b) (4 points) Is the metric space (X, doo) complete? (Prove your answer).

Problem 2. 7pts.

Let

=

=

X := {(an)n>1 CR; ƏN E N such that V n >N: An = 0},

and let doo(lan)n21, (bn)n>1) := supn>1 an – Enl.

.

(a) (3 points) Prove that (X, doo) is a metric space.

(b) (4 points) Is the metric space (X, doo) complete? (Prove your answer).

Problem 3. 7pts.

i.) (4 points) Let (X, d) be a metric space, and let K1, K2, …, Kn be

a finite family of compact subsets of X. Proved that 0–1 Ki is a

compact set.

ii.) (3 points) Let (X, d) be a compact metric space. Prove or give a

counterexample: If Ki C K2 C K3 C… is an increasing sequence

of compact subsets of X. Then Um=1Kn is compact.

n

Problem 4. 9pts.

i.) (4 points) Let (X, d) be a metric space. Suppose (2n)nen is a sequence

in X which converges to x E X. Let S := {Xn; n E N} U {x}. Show

that S is compact. (Note: You can use any of the characterizations

of compactness from class).

ii.) (5 points) Let (X, d) be a metric space, let K CX be compact and

let C C X be closed. Assume that

inf{d(x, y); x E K,Y EC} = 0.

х

Prove that KNC +0.

Purchase answer to see full

attachment

Explanation & Answer:

4 Questions

Tags:

mathematics

metric space

sequence

User generated content is uploaded by users for the purposes of learning and should be used following Studypool’s honor code & terms of service.

## Reviews, comments, and love from our customers and community:

This page is having a slideshow that uses Javascript. Your browser either doesn't support Javascript or you have it turned off. To see this page as it is meant to appear please use a Javascript enabled browser.