Using cases in proofs

**fruitcakelover** · September 2nd 2007, 06:01 AM

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.

**ThePerfectHacker** · September 2nd 2007, 06:38 AM

Originally Posted by fruitcakelover

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.

Given $x\in \mathbb{R}$ we define $|x| = x \mbox{ if }x\geq 0 \mbox{ and }|x| = -x\mbox{ if }x<0$ .

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

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

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