Suppose we have the induced matrix norm for an arbitrary $n\times n$ matrix $A$ given by,
\begin{equation*} |||A|||_{\infty,h} = \max_{1\leq i \leq N}\left(\sum_{j=1}^N |A_{ij}|\right). \end{equation*}
where $h = \frac{1}{N+1}$ and $i=1,2,3....N$ and $j=1,2,3...M$ and $N, M \in \mathbb{N}$. Additionally, suppose we have the vector $U$ given by, \begin{equation*} U^j = \begin{pmatrix} u_1^j\\ u^j_2\\ \vdots\\ u^j_N \end{pmatrix}~\in\mathbb{R}^N. \end{equation*} where $u_i^j$ is just the discrete solution to a governing problem. Additionally, we define the infinity norm as such, \begin{equation*} ||U||_{\infty,h} = \max_{1\leq i \leq N} |U_i|. \end{equation*}
The following inequality is true, \begin{equation*} ||AU||_{\infty,h}\leq|||A|||_{\infty,h} ||U||_{\infty,h}, \end{equation*} however, I have no idea how one can obtain this result.