# Thread: complex analysis

1. ## complex analysis

Hi, How would you show that for all z,w in the complex no. that ||z|-|w|| is less than or equal to |z-w| and that if the sequence z_n tends to w it implies that |z_n| tends to |w|.

They probably follow a similar pattern to the proof for real numbers but I'm unsure about bits of it.

It also asks how to show that for theta in the reals, ||z|-|w|| is less than or equal to |z+we^(itheta)| and hence that ||z|-|w|| = min{|z +we^(itheta)| : theta in reals}

I'm not even sure if we've done this in class.

thanks for any help

2. Originally Posted by tescotime
How would you show that for all z,w in the complex no. that ||z|-|w|| is less than or equal to |z-w| and that if the sequence z_n tends to w it implies that |z_n| tends to |w|.
Assuming that you know the triangle inequality $\displaystyle |u+v|\leqslant|u|+|v|$, all you have to do is to apply it with u=z-w and v=w. Then apply it again with u=w-z and v=z.

The second part then follows easily from that result, because $\displaystyle ||z_n|-|z||\leq|z_n-z|.$