TITLE: Complex number question on proving an inequality.
QUESTION [6 upvotes]: If $|z_1|=1,|z_2|=1$, how can one prove $|1+z_1|+|1+z_2|+|1+z_1z_2|\ge2$

REPLY [4 votes]: $$ \mid 1+z_1 \mid + \mid 1+ z_2 \mid + \mid 1+ z_1z_2 \mid \  \ge \mid 1+z_1 \mid + \mid 1+z_1z_2-1-z_2 \mid$$ [ Using Triangle inequality]
$$  \mid 1+z_1 \mid + \mid z_1z_2-z_2 \mid =  \mid 1+z_1 \mid + \mid z_1 -1 \mid  \ \ge |1+z_1+z_1-1| = 2  $$ [Again using triangle inequality]
So we are done :)