I want to prove $ x\leq \lvert x \rvert$.
We have two cases:
If $x\geq 0$: We have $x \leq \lvert x \rvert$ (or should it be $x \leq \lvert x \rvert=x$ ?)
If $x<0$: We have $-x<- \lvert x \rvert \iff x> \lvert x \rvert$ (Or maybe $x<- \lvert x \rvert \iff -x> \lvert x \rvert$ ?)
So I have $x\leq \lvert x \rvert$ and $x>\lvert x\rvert$, but now what?
When I check the two cases, should I only change the sign for the $x$ in absolute value-sign or also the normal $x$ (case 2)?
Thanks!