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Homework Help: Does triangle inequality hold for summations and sup?

  1. Sep 5, 2005 #1
    I know that the triangle inequality is lx+y|<= |x| +|y|

    Does this hold for a summation of say Sigma (|xn+yn|) <= sigma|xn| + sigma|yn| for n=0 to infinity?

    would this also work for sup|x+y| ??
     
  2. jcsd
  3. Sep 5, 2005 #2

    LeonhardEuler

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    Gold Member

    Yes:
    [tex]\sum_{n=0}^{N}{|x_n+y_n|}[/tex]
    [tex]\leq |x_0| + |y_0| + \sum_{n=1}^{N}{|x_{n}+y_n|}[/tex]
    [tex]\leq |x_0| + |y_0| + |x_1| + |y_1| + \sum_{n=2}^{N}{|x_{n}+y_n|}[/tex]
    ...
    [tex]\leq \sum_{n=0}^{N}{|x_n|} +\sum_{n=0}^{N}{|y_n|}[/tex]
     
    Last edited: Sep 5, 2005
  4. Sep 5, 2005 #3

    AKG

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    LeonhardEuler, the original poster asked about infinite sums, not sums to N (which I assume you used to represent some natural number). If the sum is infinite, then if xn = 1 for all n, and yn = -1 for all n, then [itex]\sum _{n=0} ^{\infty} |x_n|[/itex] and [itex]\sum _{n=0} ^{\infty} |y_n|[/itex] aren't even defined.

    Suppose all the series in question do in fact converge. Then suppose the desired inequality is not true, then we'd have:

    [tex]\sum _{n = 0} ^{\infty} |x_n + y_n| > \sum _{n=0} ^{\infty} |x_n| + \sum _{n=0} ^{\infty} |y_n|[/tex]

    [tex]\lim _{N \to \infty} \left ( \sum _{n = 0} ^N |x_n + y_n|\right ) > \lim _{N \to \infty} \left ( \sum _{n = 0} ^N |x_n|\right ) + \lim _{N \to \infty} \left ( \sum _{n = 0} ^N |y_n|\right )[/tex]

    [tex]\lim _{N \to \infty} \left ( \sum _{n = 0} ^N |x_n + y_n|\right ) > \lim _{N \to \infty} \left ( \sum _{n = 0} ^N |x_n| + \sum _{n = 0} ^N |y_n|\right )[/tex]

    thus there is some N such that:

    [tex]\sum _{n = 0} ^N |x_n + y_n| > \sum _{n = 0} ^N |x_n|\right + \sum _{n = 0} ^N |y_n|[/tex]

    which LeonhardEuler has proven false.
     
  5. Sep 5, 2005 #4
    alright thanks guys!
     
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