I've been trying to prove that the following is a norm, but wasn't successful. I also cannot find a counterexample. So help is greatly appreciated. Let $x \in \mathbb{R}^N, \ w_i \in \mathbb{R}_+,\ i=1,\ldots,N$. $\lVert x \rVert_w := \max \lvert w_i x_i\rvert$
Basically, this is the maximum norm with positive weights assigned to each dimension.
It must be shown that: $\max \lvert w_i (x_i+y_i) \rvert \leq \max \lvert w_j x_j \rvert + \max \lvert w_k y_k \rvert$