Results 1 to 7 of 7

Math Help - Equivalent Metrics

  1. #1
    Junior Member
    Joined
    Sep 2009
    From
    Johannesburg, South Africa
    Posts
    71

    Equivalent Metrics

    Define \rho on X \times X by

    \rho(x,y) = min(1,d(x,y)), \ \ \ \ \ \  \ \ \ \ x,y \in X

    Show that \rho is a metric that is equivalent to d

    My solution:
    Two metrics are equivalent if and only if the convergent sequences in (X,d) are the same as the convergent sequences in (X,\rho)

    Let (x_{n})^\infty_{n=1} be convergent in \rho i.e.,

    \forall \  \epsilon \  \exists \  N_{\epsilon}\  \mbox{such that}\  \forall\ n \geq N_{\epsilon}\  \rho(x_n,x)={min(1,d(x_{n},x))}}<\epsilon

    Choose 0 \ < \epsilon \ < 1. Then \exists \ N_{\epsilon} such that \forall\ n \ \geq N_{\epsilon}: min (1,d(x_n,x)) < \epsilon < 1.

    This implies min (1,d(x_n,x))=d(x_n,x)\ <\ \epsilon.

    What if \epsilon > 1?
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor

    Joined
    Apr 2005
    Posts
    14,973
    Thanks
    1121
    Irrelevant. The definition of "convergent" says for any \epsilon> 0 but is is small \epsilon that is important.
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Super Member
    Joined
    Aug 2009
    From
    Israel
    Posts
    976
    And if you still wish to show it for \epsilon>1, you can simply note that min(x,y) \le x &  min(x,y) \le y \ \forall x,y \in \mathbb{R}
    Follow Math Help Forum on Facebook and Google+

  4. #4
    Moo
    Moo is offline
    A Cute Angle Moo's Avatar
    Joined
    Mar 2008
    From
    P(I'm here)=1/3, P(I'm there)=t+1/3
    Posts
    5,618
    Thanks
    6
    Defunkt, how would that be relevant ? oO

    I think it's possible to show it for e>1 if you consider the contrapositive. But as Hallsofivy says, it's not useful
    Follow Math Help Forum on Facebook and Google+

  5. #5
    Super Member
    Joined
    Aug 2009
    From
    Israel
    Posts
    976
    min(1, d(x_n,x)) \le 1 < \epsilon
    Follow Math Help Forum on Facebook and Google+

  6. #6
    Moo
    Moo is offline
    A Cute Angle Moo's Avatar
    Joined
    Mar 2008
    From
    P(I'm here)=1/3, P(I'm there)=t+1/3
    Posts
    5,618
    Thanks
    6
    But what we want to show is that d(x_n,x)<\epsilon. We already know that \min(1,d(x_n,x))<\epsilon
    Follow Math Help Forum on Facebook and Google+

  7. #7
    Super Member
    Joined
    Aug 2009
    From
    Israel
    Posts
    976


    Oops
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Fun with metrics
    Posted in the Differential Geometry Forum
    Replies: 4
    Last Post: September 24th 2011, 07:05 AM
  2. Are equivalent metrics comparable on compact sets?
    Posted in the Differential Geometry Forum
    Replies: 2
    Last Post: August 18th 2010, 07:23 AM
  3. Equivalent Metrics
    Posted in the Differential Geometry Forum
    Replies: 10
    Last Post: February 16th 2010, 03:24 AM
  4. Challenging Problem!!! (Equivalent Metrics)
    Posted in the Differential Geometry Forum
    Replies: 13
    Last Post: January 23rd 2010, 10:23 AM
  5. Metrics
    Posted in the Calculus Forum
    Replies: 1
    Last Post: November 29th 2008, 11:38 AM

Search Tags


/mathhelpforum @mathhelpforum