local maximum principle for monotone schemes

Question

Consider a monotone scheme $$ u_{k}^{n+1}=G\left(u_{k-p-1}^{n}, \ldots, u_{k+q}^{n}\right) $$ We have that if $u_{k} \leq v_{k}$ for all $k,$ then $G(u)_{k} \leq G(v)_{k}$ for all $k$. How can we show that this monotone scheme satisfies local maximum principle: $$ \text {min}_{i \in \text {stencil around} k} u_{j} \leq G(u)_{j} \leq m a x_{i \in \text { stencil around } j} u_{i} $$ where we take $$ v_{i}=\left\{\begin{array}{ll}{\max _{k \in s t e n c i l \text { around } i} u_{k},} & {\text { if } k \in \text { stencil around } i} \\ {u_{i},} & {\text { otherwise }}\end{array}\right. $$