I need to prove $|z-1| \le ||z|-1|+|z||\arg z|$ using a purely geometric argument.
I know that $|z-1|$ is the distance from z to (1,0). $|z|-1$ is the distance from $z$ to the edge of the unit circle when $z$ is outside of the unit circle. (Similarly, $1-|z|$ is the distance from $z$ to the edge of the unit circle when $z$ is inside of the unit circle.)
I am, however, having a bit of difficulty interpreting what $|z||\arg z|$ is geometrically.
I know that the two vectors $|z-1|$ and $|z|-1$ (or $1-|z|$) seem to form 2 sides of a triangle. I think that $|z||\arg z|$ must somehow be related to that last side of the triangle.
Is $|z||\arg z|$ related to an arc? If so, how is that related to the triangle listed above?
Thanks for all your help in advance!