Applying R1557: Weighted real AM-GM inequality to the real numbers $y_n : = x^{\lambda_n}_n$ with weights $\theta_n : = 1 / \lambda_n$, we have \begin{equation} \prod_{n = 1}^N x_n = \prod_{n = 1}^N (x^{\lambda_n}_n)^{1 / \lambda_n} = \prod_{n = 1}^N y_n^{\theta_n} \leq \sum_{n = 1}^N \theta_n y_n = \sum_{n = 1}^N \frac{1}{\lambda_n} x^{\lambda_n}_n \end{equation} This establishes the first claim. The second claim follows from the equality condition of the same result. $\square$