Hi, does anybody know how to prove the following statement:

For each real number r, -|r|£ r £ |r|.

by using two cases in the proof.

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- Sep 2nd 2007, 06:01 AMfruitcakeloverUsing cases in proofs
Hi, does anybody know how to prove the following statement:

For each real number r, -|r|£ r £ |r|.

by using two cases in the proof. - Sep 2nd 2007, 06:38 AMThePerfectHacker
Given $\displaystyle x\in \mathbb{R}$ we define $\displaystyle |x| = x \mbox{ if }x\geq 0 \mbox{ and }|x| = -x\mbox{ if }x<0$.

Case 1, when $\displaystyle x\geq 0$. Thus $\displaystyle |x|=x$. And so we need to prove $\displaystyle -x\leq x\leq x$. Which is true since $\displaystyle x\leq x$ and $\displaystyle -x<0\leq x$.

Case 2, when $\displaystyle x<0$. You continue.