On an inequality between Dirichlet and Neumann eigenvalues for the Laplace operator

A simple proof of the inequality $\mu_{k+1} < \lambda_k$ is given. Here the $\lambda_k$ (respectively, $\mu_k$) are the eigenvalues of the Dirichlet (respectively, Neumann) problem for the Laplace operator in an arbitrary domain of finite measure in $\mathbb{R} ^d$, $d>1$.