Exercise 4.3.1 (Basic properties of absolute value and distance)

Prove Proposition 4.3.3.

Proposition 4.3.3 Let x,y,z be rational numbers.

(a) (Non-degeneracy of absolute value) We have $\left|x\right|\ge 0$. Also, $\left|x\right|=0$ if and only if $x$ is $0$.
(b) (Triangle inequality for absolute value) We have $|x+y|\le \left|x\right|+\left|y\right|$.
(c) We have the inequalities $-y\le x\le y$ if and only if $y\ge \left|x\right|$. In particular, we have $-\left|x\right|\le x\le \left|x\right|$.
(d) (Multiplicativity of absolute value) We have $\left|\mathit{xy}\right|=\left|x\right|\left|y\right|$. In particular, $|-x|=\left|x\right|$.
(e) (Non-degeneracy of distance) We have $d(x,y)\ge 0$. Also, $d(x,y)=0$ if and only if $x=y$.
(f) (Symmetry of distance) $d(x,y)=d(y,x)$.
(g) (Triangle inequality for distance) $d(x,z)\le d(x,y)+d(y,z)$.