Let $(x, y, u, v) \in \left(\mathcal{D}_{\mathbf{N}}\right)^4$. Show that, for all natural number $k$, $$|(x * y)(k) - (u * v)(k)| \leq \sum_{i+j=k} y(j)|x(i) - u(i)| + \sum_{i+j=k} u(i)|y(j) - v(j)|.$$
Let $(x, y, u, v) \in \left(\mathcal{D}_{\mathbf{N}}\right)^4$. Show that, for all natural number $k$,
$$|(x * y)(k) - (u * v)(k)| \leq \sum_{i+j=k} y(j)|x(i) - u(i)| + \sum_{i+j=k} u(i)|y(j) - v(j)|.$$