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Homework Help: Generalized triangle inequality

  1. Aug 15, 2012 #1
    1. The problem statement, all variables and given/known data

    Show that

    |x_1 + x_2 + · · · + x_n | ≤ |x_1 | + |x_2 | + · · · + |x_n |

    for any numbers x_1 , x_2 , . . . , x_n


    2. Relevant equations

    |x_1 + x_2| ≤ |x_1| + |x_2| (Triangle inequality)


    3. The attempt at a solution

    I tried using the principle of induction here, but to no avail.

    Can I induct on the basis |x_1| ≤ |x_1| ?
     
  2. jcsd
  3. Aug 15, 2012 #2

    gabbagabbahey

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    Hint: [itex]| x_1+x_2 + x_3 | \leq |x_1 + x_2| + | x_3 |[/itex]
     
  4. Aug 15, 2012 #3
    so I can write |x_1 + x_2 + x_3| ≤ |x_1| + |x_2| + |x_3|

    since |x_1 + x_2| ≤ |x_1| + |x_2|

    but how do I cover all the "n" cases?
     
  5. Aug 15, 2012 #4

    gabbagabbahey

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    Use the inductive principle... assume that [itex]| \sum_{i=1}^n x_i| \leq \sum_{i=1}^n |x_i|[/itex] for some some [itex]n=k[/itex] (it is obviously true for n=1, 2 and 3) , and then show that it must then also be true for [itex]n=k + 1[/itex].
     
  6. Aug 15, 2012 #5

    Ray Vickson

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    You can use the easily-proven fact that the absolute-value function is convex, in the sense that f(x) satisfies [itex]f(\alpha w_1 + (1-\alpha)w_2) \leq \alpha f(w_1) + (1-\alpha) f(w_2)[/itex] for all [itex] \alpha \in [0,1].[/itex] Try to prove that
    [tex] \left| \frac{x_1 + x_2 + \cdots + x_n}{n}\right| \leq \frac{1}{n}|x_1| + \cdots + \frac{1}{n} |x_n|.[/tex] Hint: induction.

    RGV
     
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