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Real Analysis triangle inequity

  1. Oct 19, 2013 #1
    Prove |x|+|y| ≤ |x+y| + |x-y| for all real numbers x and y.
    Some ideas I have is let a = x+y and b = x-y and apply triangle inequity
    Could anyone give me some direction?

  2. jcsd
  3. Oct 19, 2013 #2

    Ray Vickson

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    Sure: just use your ideas above and see where they lead.
  4. Oct 19, 2013 #3
    let a = x+y and b = x-y.
    |2x| = |a+b| <= |a| + |b|
    |2y| = |a- b| <= |a| + |b|
    So |2x| + |2y| <= 2(|a| + |b|)
    Divide both sides by 2, we get
    |x|+ |y| <= |a| + |b|

    Is this right way?

  5. Oct 19, 2013 #4


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    Sure, that's a fine proof.
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