Thread: Triangle Inequality

I am trying to understand the proof without using the square/square root function. From what I understand there are two equivalent statements i.e. |p+q| <= |p|+|q| and | |p|-|q| | <= |p-q|

The first one I can derive. It is the second one that I am having trouble with. Can someone please show me the steps from the first to the second?
Thank you.

Just use the first one in the form with the assumption that . If this isn't the case then just relabel since and