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 $\displaystyle x\in \mathbb{R}$ 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.