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Verifying the metric space e = d / (1 + d)

  1. Sep 5, 2011 #1
    I'm trying to verify that if (M,d) is a metric space, then (N,e) is a metric space where e(a,b) = d(a,b) / (1 + d(a,b)). Everything was easy to verify except the triangle inequality. All I need is to show that:

    a <= b + c
    implies
    a / (1 + a) <= (b / (1 + b)) + (c / (1 + c)

    Any help would be greatly appreciated, thank you!
     
  2. jcsd
  3. Sep 5, 2011 #2

    tiny-tim

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    hi pdonovan! :smile:

    try writing everything as a fraction over (1 + a)(1 + b)(1 + c), and seeing what cancels :wink:
     
  4. Sep 5, 2011 #3
    Thank you for the tip, but I still haven't figured it out : /

    I have
    a<=b+c
    -->
    a/[(1+a)(1+b)(1+c)] <= (b+c)/[(1+a)(1+b)(1+c)]
    -->
    a/[(1+a)(1+b)(1+c)] <= b/[(1+a)(1+b)(1+c)] + c/[(1+a)(1+b)(1+c)]
     
  5. Sep 6, 2011 #4

    tiny-tim

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    (just got up :zzz: …)

    no, start at the answer, and work backwards! :rolleyes:
     
  6. Sep 6, 2011 #5
    like tiny-tim said, write down the triangle inequality for the new metric and assume it's true. then multiply both sides by (1+a)(1+b)(1+c) (since it's positive) and then "work backwards" as tiny-tim said. then see if you can reversely do all the steps from the opposite direction. if you can, (and you can), then you're done.
     
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