An easy difficult problem

  • Thread starter evagelos
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  • #1
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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??
 

Answers and Replies

  • #2
CompuChip
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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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