Regarding the derivation of triangle inequality related inequality (undergraduate complex analysis) I am using Brown and Churchill's Complex Analysis Textbook, and on pg.11 of the eighth edition, there is a triangle inequality derivation as followed
to prove 
$|z_1+z_2|\geq ||z_1|-|z_2||$
$|z_1|=|(z_1+z_2-z_2)|\leq|z_1+z_2|+|z_2|$  
therfore
$|z_1+z_2|\geq|z_1|-|z_2|$
however I realized that if 
$-|z_1|=-|(z_1+z_2-z_2)|\leq-(|z_1+z_2|+|z_2|)$  
then
$|z_1+z_2|\leq|z_1|-|z_2|$
which is a contradiction
does staring on $|z_1|$ implies an inherent assumption I have to made, if so what is the assumption?
moreover, when they reached the 
$|z_1+z_2|\geq|z_1|-|z_2|$
conclusion, they say this only work for when $|z_1|\geq|z_2|$
when $|z_1|<|z_2|$ they exchange $z_1$ and $z_2$ to arrive $|z_1+z_2|\geq-1*(|z_1|-|z_2|)$ for which I don't follow as why would we care about the order of $z_1$ and $z_2$, isn't the designation of $z_1$ and $z_2$ arbitrary?
Any hint would be much appreciated
 A: First of all, let's recall that multiplying by a negative number switches inequality signs.  That is, $x \leq y$ does not imply $-x \leq -y$, but rather $-x \geq -y$.  In particular, your inequality
$$-|z_1 + z_2 - z_2| \leq -(|z_1 + z_2| + |z_2|)$$
is generally false.  For example, if you take $z_1 = z_2 = 1$, you get that $-1 \leq -3$, which is incorrect.
When the authors reach the point that $|z_1 + z_2| \geq |z_1| - |z_2|$, the proof is not yet finished.  Remember that the point is to show that $|z_1 + z_2| \geq ||z_1| - |z_2||$.  To get those extra absolute value bars we need an additional step.  So:
If it's the case that $|z_1| \geq |z_2|$, then we have $|z_1| - |z_2| \geq 0$, and so $||z_1| - |z_2|| = |z_1| - |z_2|$.  So in this case, we really are done because we've shown
$$|z_1 + z_2| \geq |z_1| - |z_2| = ||z_1| - |z_2||.$$
But what if we don't have $|z_1| \geq |z_2|$?  Well, then we must have $|z_1| < |z_2|$.  But this is just like the previous case, but with the roles of $z_1$ and $z_2$ interchanged, meaning that we have
$$|z_2 + z_1| \geq |z_2| - |z_1| = ||z_2| - |z_1||.$$
Since $|z_1 + z_2| = |z_2 + z_1|$ and $||z_1| - |z_2|| = ||z_2| - |z_1||$, we're done.
