# Thread: Using cases in proofs

1. ## Using 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.

2. 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.