I have the following question. Let $I$ be a finite set, $\{S_i\}_{i \in I}$ be a family of sets and $S = \prod_{i \in I} S_i$ the Carthesian product of this family of sets. Consider the bounded function $\pi_i: S \rightarrow \mathbb{R}$. Denote by $(h_i, s_{-i})$ the point (vector) in $S$ where we replace the element $s_i$ by $h_i$. I am trying to prove that the following inequality holds true:
\begin{align} \sup_{i \in I, h_i \in S_i} [\pi_i (h_i, t_{-i}) - \pi_i(t)] \leq \sup_{i \in I, h_i \in S_i}[ \pi_i (h_i, t_{-i}) - \pi_i(s)] + p(s,t) \end{align}
where $$p(s,t) = \sum_{i \in I} \sup_{h_{-i} \in S_{-i}} \max_{j \in I} |\pi_j(s_i, h_{-i}) - \pi_j(t_i, h_{-i})|$$