Showing infinity norm of diagonally dominant matrix is less than 1

Question

We have a matrix A that is diagonally dominant by rows with decomposition $A = D - L - U$. The matrix $T = D^{-1}(L+U)$ and I want to show that $\lVert T\rVert_\infty < 1$ is always true. I know that $D^{-1}$ will always be a diagonal matrix where all the numbers are $ \leq 1$, meaning when we multiply it by $(L+U)$ the resulting matrix will have numbers all $ <1$ and thus the infinity norm will also be $<1$. However I am unsure of how to write this logic out mathematically.