Show that $\sup_{t \in T}(t_1+\phi(t_2))+\sup_{t\in T} (t_1-\phi(t_2))\le\sup_{t \in T}(t_1+t_2)+\sup_{t\in T} (t_1-t_2)$ where $\phi$ is a contraction (Lipschitz with Lipschitz constant less than 1) and $T \subset \mathbb{R^2}$ is bounded.
There was a solution posted here An inequality about suprema and contraction function. But I think the soution is false there. I have thought about this exercise long and hard. It seems that I'm missing something obvious.