$\begin{align*} |x|=x &\text{if }x\geq 0\\ |x|=-x &\text{if }x\lt 0. \end{align*}$ Show that $|xy|=|x||y|$.
I try to prove it as follows:
$|xy|=xy$ if $xy\geq 0$, but $xy\geq 0$ if and only if $x\geq 0$ and $y\geq 0$. Since $x\geq 0$ and $y\geq 0$, then $x=|x|$ and $y=|y|$ hence from $|xy|=xy$ we have $|xy|=|x||y|$.
Am I correct?
How can I show the following two cases?
i. Show that $|x+y|\leq |x|+|y|$.
ii. If $y\gt 0$ and $-y\leq x \leq y$, then $|x|\lt y$