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How can I use the order axioms of $\mathbb{R}$ to prove the semi-definite positivity property for the absolute value:

For all $x \in \mathbb{R}, |x|\geq0$ and $|x|=0$ if and only if $x=0$?

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  • $\begingroup$ Hi Jake! What have you tried so far? $\endgroup$ – user458276 Feb 5 at 16:12
  • $\begingroup$ @user458276 I have tried proving by contradiction. So, suppose $|x|\geq0$ and $|x|=0$ and $x\neq0$. Obviously, we have a contradiction because $x$ cannot both be 0 and not 0. I just don't see the motivation behind using the order axioms of the reals. $\endgroup$ – Jake Feb 5 at 16:15
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If 0 <= x, then -x <= 0. Thus |x| = max(x,-x) = x >= 0.
If x <= 0, then 0 <= -x. Thus |x| = max(x,-x) = -x >= 0.

If |x| = 0, then x = 0 or -x = 0. Thus x = 0.
If x = 0, then |x| = 0.

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