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Proof of square root properties

  1. Sep 4, 2011 #1
    1. The problem statement, all variables and given/known data
    [itex]\sqrt{\sum}x^{2}_{i}[/itex][itex]\leq[/itex][itex]\sum[/itex]|x[itex]_{i}[/itex]|[itex]\leq[/itex][itex]\sqrt{n}[/itex][itex]\sqrt{\sum}x^{2}_{i}[/itex]

    *The sums are all from i=1 to n.*


    2. Relevant equations



    3. The attempt at a solution
    I'm very new to proof-based math, and just looking for some help to get started with this one. Thanks in advance.
     
  2. jcsd
  3. Sep 4, 2011 #2

    lanedance

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    For this part
    [tex] \sqrt{\Sum_i x_i^2} \leq \Sum_i |x_i|[/tex]

    it should be clear
    [tex] 0 \leq \sqrt{\Sum_i x_i^2} [/tex]
    [tex] 0 \leq \leq \Sum_i |x_i|[/tex]

    so squaring both sides could be useful
     
  4. Sep 6, 2011 #3
    I tried that and didn't get far. It seems to my the leftmost inequality is always equal. I must be thinking about it wrong. In what instance would that inequality be less than?
     
  5. Sep 7, 2011 #4

    lanedance

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    you should some cross terms like |xi||xj| on in the middle, which don't appear on the left
     
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