Results 1 to 4 of 4

Math Help - Inequality in Metric spaces

  1. #1
    Newbie
    Joined
    Jul 2011
    From
    .
    Posts
    17

    Inequality in Metric spaces

    Let (X,d) be a metric space. Prove that for all w,x,y,z \in X:

    |d(w,x)-d(y,z)|\le d(w,y)+d(x,z)

    I haven't been able to prove it, I'm sure the triangle inequality should be used but I don't know how. Thanks in advance.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor Also sprach Zarathustra's Avatar
    Joined
    Dec 2009
    From
    Russia
    Posts
    1,506
    Thanks
    1

    Re: Inequality in Metric spaces

    Quote Originally Posted by MATHNEM View Post
    Let (X,d) be a metric space. Prove that for all w,x,y,z \in X:

    |d(w,x)-d(y,z)|\le d(w,y)+d(x,z)

    I haven't been able to prove it, I'm sure the triangle inequality should be used but I don't know how. Thanks in advance.

    To prove:

    |d(w,x)-d(y,z)|\le d(w,y)+d(x,z)

    you need to prove:

    1. d(w,x)-d(y,z) \le d(w,y)+d(x,z)

    2. d(w,x)-d(y,z)\geq  -(d(w,y)+d(x,z))
    Last edited by Also sprach Zarathustra; January 7th 2012 at 11:35 PM.
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Newbie
    Joined
    Jul 2011
    From
    .
    Posts
    17

    Re: Inequality in Metric spaces

    Quote Originally Posted by Also sprach Zarathustra View Post
    To prove:

    |d(w,x)-d(y,z)|\le d(w,y)+d(x,z)

    you need to prove:

    1. d(w,x)-d(y,z) \le d(w,y)+d(x,z)

    2. d(w,x)-d(y,z)\geq  -(d(w,y)+d(x,z))
    I hadn't seen the inequality that way Thank you so much.
    Follow Math Help Forum on Facebook and Google+

  4. #4
    MHF Contributor

    Joined
    Aug 2006
    Posts
    18,968
    Thanks
    1785
    Awards
    1

    Re: Inequality in Metric spaces

    Quote Originally Posted by MATHNEM View Post
    Let (X,d) be a metric space. Prove that for all w,x,y,z \in X:
    |d(w,x)-d(y,z)|\le d(w,y)+d(x,z)
    Start with
    \begin{align*}d(w,x) &\le d(w,y)+d(y,x)\\ d(w,x)-d(x,y) &\le d(w,y)    \end{align*}

    Again
    \begin{align*}d(x,y) &\le d(x,z)+d(z,y)\\ d(x,y)-d(z,y) &\le d(x,z)    \end{align*}.

    Now add.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Any metric spaces can be viewed as a subset of normed spaces
    Posted in the Differential Geometry Forum
    Replies: 5
    Last Post: December 15th 2011, 04:00 PM
  2. Metric Spaces
    Posted in the Math Challenge Problems Forum
    Replies: 1
    Last Post: January 23rd 2010, 08:42 AM
  3. metric spaces
    Posted in the Differential Geometry Forum
    Replies: 1
    Last Post: May 3rd 2009, 01:33 AM
  4. More metric spaces
    Posted in the Advanced Algebra Forum
    Replies: 1
    Last Post: October 15th 2008, 12:44 PM
  5. Metric Spaces
    Posted in the Advanced Algebra Forum
    Replies: 2
    Last Post: November 26th 2007, 10:11 PM

Search Tags


/mathhelpforum @mathhelpforum