Let \(\Omega\subseteq \mathbb{R}^d\) be bounded and \(u, v \in C^0(\Omega)\), with \(u\) subharmonic and \(v\) superharmonic on \(\Omega\).
If \(u \le v\) on \(\partial \Omega\), then \(u\le v\) in \(\Omega\).
Proof
We apply the strong maximum principle
on \(u - v\), which is subharmonic.