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How do you prove this consequence of the triangle inequality?

  1. Apr 16, 2009 #1
    I should know how to do this, but I just can't figure it out. Should be a piece of cake. How do you prove, for [itex]x,y\in \mathbb{R}^n[/itex]:

    [tex]
    \left| (||x|| - ||y||) \right| \leq ||x-y||
    [/tex]
     
  2. jcsd
  3. Apr 16, 2009 #2

    arildno

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    We have:
    [tex]||x+y||\leq{||}x||+||y||[/tex]

    Now, set x=z-y

    Then, we get:
    [tex]||z||\leq{||z-y||}+||y||[/tex],
    For completely arbitrary z and y.

    utilize this to derive the desired relation, i.e:
    [tex]-||u-v||\leq{||u||}-||v||\leq{||}||u-v||[/tex]
     
  4. Apr 16, 2009 #3
    ||x||=||x-y+y||[tex]\leq[/tex]||x-y||+||y|| ======>

    ||x||-||y||[tex]\leq[/tex]||x-y||............................................................1

    ||y||= ||y-x+x||[tex]\leq[/tex]||x-y||+||x||======>

    ||y||-||x||[tex]\leq||x-y||[/tex] =======>

    ||x||-||y||[tex]\geq[/tex]-||x-y||............................................................2

    from (1) and (2) we get:


    -||x-y||[tex]\leq ||x||-||y||\leq ||x-y|| \Longleftrightarrow|(||x||-||y||)|\leq||x-y||[/tex]
     
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