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!