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An easy difficult problem

  1. May 31, 2010 #1
    Every body knows that:

    [tex]x_{1}^2+x_{2}^2........x_{n}^2 =0\Longrightarrow x_{1}=0\wedge x_{2}=0........\wedge x_{n}=0[/tex].

    But how do we prove that?

    Perhaps by using induction?

    For n=1 .o.k

    Assume true for n=k

    And here now is the difficult part .How do we prove the implication for n=k+1??
  2. jcsd
  3. Jun 1, 2010 #2


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    Induction sounds like a bit of overkill here, but if you insist...
    Of course it is true that if
    [tex]a + b = 0[/tex]
    then either a = b = 0, or a = -b (not equal to 0).
    You can use this for
    [tex]a = x_1^2 + x_2^2 + \cdots + x_n^2, \quad b = x_{n + 1}^2[/tex]
    and use that [itex]x_i^2 \ge 0[/itex] for all i.
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