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Metric space Triangular inequalities

  1. Feb 10, 2010 #1
    1. The problem statement, all variables and given/known data

    let [tex]
    (X,\sigma) [/tex] be a metric space. [tex] xyz \in R[/tex]show that
    [tex]
    \mid \sigma(x,z)-\sigma(y,z) \mid \leq \sigma(x,y)[/tex]

    2. Relevant equations



    3. The attempt at a solution

    [tex]\mid \sigma(x,z)-\sigma(y,z) \mid \leq \sigma(x,y)=\mid \sigma(x,y) \mid = \mid\sigma
    (z,x) + \sigma(z,y) \mid[/tex]

    [tex] \leq \mid\sigma
    (x,z)\mid + \mid \sigma(y,z) \mid[/tex]


    Does that look alright?
    1. The problem statement, all variables and given/known data



    2. Relevant equations



    3. The attempt at a solution
     
  2. jcsd
  3. Feb 11, 2010 #2
    For real numbers a and b, |a+b| [tex]

    \leq
    [/tex] |a| + |b|.
    Thus, |a| + |b|.= |(a-c) + (c-b)| [tex]

    \leq
    [/tex] |a-c| + |c-b|
     
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