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
however I realized that if
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
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