Let $n = 1, \ldots, N$. Using result R5188: and multiplying both sides by $\lambda_n$, we have the inequality \begin{equation} \lambda_n x_n y_n \leq \frac{\alpha^p}{p} \lambda_n x^p_n + \frac{1}{q \alpha^q} \lambda_n y^q_n \end{equation} with equality if and only if $\frac{\alpha^p}{p} x^p_n = \frac{1}{q \alpha^q} y^q_n$. Summing both sides over $n = 1, \ldots, N$ now yields the claim. $\square$