I need to prove the following. Set
$$|\cdot|_1: R^n \rightarrow R, \;\; |x|_1 = \sum_{i=1}^n |x(i)|$$
and
$$X = \{x \in R^n|\, 0 \leq x(i) \leq 1\; \forall i\}$$
Then $$|z|_1|x-y|_1( |x|_1 +\ |y|_1)\ \leq\ |z|_1\ (|x-z|_1|y|_1+\ |y-z|_1|x|_1) +\ |x-z|_1|y-z|_1(|x|_1+\ |y_1|) +\ |x|_1|y|_1|x-y|_1$$
if $|x|_1 \leq |y|_1 \leq |z|_1$.
It's not homework, so I am not sure whether it is actually true or not. But I tested millions of vectors in different dimensions with my computer, so it might be true. This is only the last part of what I was working on today and I don't want to start from scratch tomorrow. I am trying to show that some weird metric satisfies the triangle inequality. But I am really stuck now. Every time I change something at the inequality above, my computer says no (i.e. he finds a counterexample).
I don't expect anyone to do my work, but maybe someone has a good idea how to continue. Maybe there is something really simple I just didn't see. Or maybe someone is able to find a counterexample. Thanks anyway :)
EDIT: I can add some code for testing, if it helps. But someone has to explain me how to insert code in stackexchange posts before.