Triangle inequality in Rubins book

[mynameisfunk]

My problem states:
Given z, w\inC, prove: ||z|-|w||\leq|z-w|\leq|z|+|w|.

Now, I am confused because, isn't it true that ||z|-|w||=|z-w| ? I am using Rudin's book which gives |z|=([z's conjugate]z)^1/2

 

[xepma]

Take z =-2 and w = 2. Does ||z|-|w||=|z-w| still hold?

 

[mynameisfunk]

Ah. Thank you. Is it correct from the definition of |z| to think of it as a length? My gf is taking a complex variables class and she had told me that |z|= the length of z, but rudin's book gave the definition above, so i didn't know how to think about it.