\sum_{k=1}^{n}|x_k+y_k|^p=\sum_{k=1}^{n}|x_k+y_k||x_k+y_k|^{p-1}\le \sum_{k=1}^{n} |x_k||x_k+y_k|^{p-1}+\sum_{k=1}^{n} |y_k||x_k+y_k|^{p-1}\le \left(\sum_{k=1}^{n}{|x_k|^p}\right)^{\frac{1}{p}}\left(\sum_{k=1}^{n}|x_k+y_k|^{q(p-1)}\right)^{\frac{1}{q}}+\left(\sum_{k=1}^{n}{|y_k|^p}\right)^{\frac{1}{p}}\left(\sum_{k=1}^{n}|x_k+y_k|^{q(p-1)}\right)^{\frac{1}{q}}=(\star)