Turn on thread page Beta
    • Thread Starter
    Offline

    0
    ReputationRep:
    hello there,
    I am desperate for help with the following question. I was absent when all this material was covered!!! The question is as follows:

    "Let d_1 be the metric on R^2 defined by d_1(x,y) = |x_1 - y_1| + |x_2 - y_2| . Prove that a sequence (x^n) [from n =1 to infinity] converges to x E R^2 in the metric space (R^2,d_1) iff both the sequence x_1^n tends to x_1 as n tends to infinity and the sequence x_2^n tends to x_2 as n tends to infinity. (Here x^n is the nth element of the sequence (x^n) [from n =1 to infinity] and x^n = (x_1^n, x_2^n) E R^2, i.e. x_1^n and x_2^n are the coordinates of x^n. "

    Any help would be more than appreciated!!! As I say, I'm desperate for help on this one!!!

    Thanks a lot for your time!!!
    Offline

    2
    ReputationRep:
    Well, let's see what we've got. Take a sequence {xi} in R2 (with xi=(xi,yi) say) and let's assume it tends to a limit x=(x,y) in R2. So, we have: given an ε>0, there exists an N such that for all n>N d1(xn,x) < ε. Then, by the definition of d1 we have:

    |x - xn| + |y - yn| < ε

    So given this ε>0, for all n>N we certainly have:

    |x - xn| < ε
    |y - yn| < ε

    So, looks like we're done in one direction. Now what about the other? Well for {xi} given an ε1>0, there exists an N1 such that for all n>N1 d(xn,x) < ε1, similarly for for {yi} given an ε2>0, there exists an N2 such that for all n>N2 d(yn,y) < ε2. Now, if we take N = max(N1,N2), ε/2 = max(ε12). Then given ε>0 for all n>N we have:

    |x - xn| + |y - yn| < ε

    And wahey - things are looking good. Hope that helped.
    • Thread Starter
    Offline

    0
    ReputationRep:
    thanks a lot for your help mate!!! And thanks for your time. Cheers, Beast.
 
 
 
Reply
Submit reply
Turn on thread page Beta
Updated: February 2, 2006

University open days

  • University of East Anglia
    All Departments Open 13:00-17:00. Find out more about our diverse range of subject areas and career progression in the Arts & Humanities, Social Sciences, Medicine & Health Sciences, and the Sciences. Postgraduate
    Wed, 30 Jan '19
  • Solent University
    Careers in maritime Undergraduate
    Sat, 2 Feb '19
  • Sheffield Hallam University
    City and Collegiate Campus Undergraduate
    Sun, 3 Feb '19
Poll
Do you have a role model?
Useful resources

Make your revision easier

Maths

Maths Forum posting guidelines

Not sure where to post? Read the updated guidelines here

Equations

How to use LaTex

Writing equations the easy way

Equations

Best calculators for A level Maths

Tips on which model to get

Student revising

Study habits of A* students

Top tips from students who have already aced their exams

Study Planner

Create your own Study Planner

Never miss a deadline again

Polling station sign

Thinking about a maths degree?

Chat with other maths applicants

Can you help? Study help unanswered threads

Groups associated with this forum:

View associated groups

The Student Room, Get Revising and Marked by Teachers are trading names of The Student Room Group Ltd.

Register Number: 04666380 (England and Wales), VAT No. 806 8067 22 Registered Office: International House, Queens Road, Brighton, BN1 3XE

Write a reply...
Reply
Hide
Reputation gems: You get these gems as you gain rep from other members for making good contributions and giving helpful advice.