Does matrix element inequality implies norm inequality?

Question

Let $A$, $B$ be two $m{\times}n$ matrices and $u > 0$. Suppose the corresponding elements from A and B satisfy \begin{equation} |B_{i,j}| \leq u | A_{i,j}| \end{equation} Can we prove \begin{equation} ||B||_1 \leq u ||A||_1 \end{equation} where $||\cdot||_1$ is the matrix norm induced by $L_1$ vector norm? In general, can we prove the following? \begin{equation} ||B||_p \leq u ||A||_p \end{equation}