Remmert chapter 1 page 13 and 14
Set $z:= x+iy, w:= u+iv$: $\langle z,w\rangle^2+\langle iz,w\rangle^2 = Re(w\bar{z})^2+Re(iw\bar{z})^2=(ux+vy)^2+(uy-vx)^2= (x^2+y^2)(u^2+v^2)= |z|^2|w|^2$
so from this it follows that: $\langle z,w\rangle^2 \le \langle z,w\rangle^2+\langle iz,w\rangle^2 = |z|^2|w|^2 \Rightarrow \langle z,w\rangle \le |z||w|$
If we put: $|z+w|^2 = |z|^2+ 2\langle z,w\rangle+ |w|^2 \le |z|^2+2|zw|+|w|^2 = (|z|+|w|)^2$
we can also say that: $|z+w| \le |z|+|w|$
lastly we can put $|z|= |z+w-w| \le |z-w|+|w| \Leftrightarrow |z|-|w| \le |z-w| \Rightarrow ||z|-|w||\le ||z-w||= |z-w|$
Have we done this correctly? Please, do tell.