Prove the triangle inequality involving complex numbers.

Our eventual goal in this problem is to prove the triangle inequality involving complex numbers.

(a) Show that for every $z ∈ C$,

$|Re(z)| ≤ |z|$ and $|Im(z)| ≤ |z|$.

(b) Given $z$, $w ∈ C$, show that

$|z+w|^2 =|z|^2 +|w|^2 +2Re(zw')$.

(c) Using parts (a) and (b), prove the triangle inequality

$|z + w| ≤ |z| + |w|$.

This is what I got.

(a) By definition for a complex number $z = x + yi$,

$$|z|^2 = x^2 + y^2 = Re(z)^2 + Im(z)^2$$

From here,

$$|z|^2 ≥ Re(z)^2 \text{ and } |z|^2 ≥ Im(z)^2$$

And, finally,

$$|z| ≥ |Re(z)| \text{ and } |z| ≥ |Im(z)|$$

(b) $|z + w|^2 = (z + w)·(z + w)'$

$$= (z + w)·[z' + w']$$

$$= zz' + [zw' + z'w] + ww'$$

$$= |z|^2 + 2Re[zw'] + |w|^2$$

$$≤ |z|^2 + 2|zw'| + |w|^2$$

$$= |z|^2 + 2|z||w| + |w|^2$$

$$= (|z| + |w|)^2.$$

(c) Since both $|z+w|$ and $|z| + |w|$ are non-negative,

$$|z + w| ≤ |z| + |w|$$

Looks solid! Sadly, I'm only seeing it well after the OP, but to cut down on unanswered questions, here we go!