How Do We Prove That All Variables Are Zero If Their Squares Sum to Zero?

[evagelos]

Every body knows that:

$$x_{1}^2+x_{2}^2...x_{n}^2 =0\Longrightarrow x_{1}=0\wedge x_{2}=0...\wedge x_{n}=0$$.


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??

 

[CompuChip]

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