Eigenelements of Neumann Laplacian satisfy $\sum_{k=1}^\infty |(u,\varphi_k)_{L^2}|^2 \lambda_k^{-\frac 12} < \infty?$

Question

Let the eigenvalues of Neumann Laplacian on a bounded open domain be given by $0 = \lambda_0 \leq \lambda_1 \leq \lambda_2 ...$ associated to eigenfunctions $\varphi_0, \varphi_1, ...$.

Let $u \in H^1$. Why is it true that $$\sum_{k=1}^\infty |(u,\varphi_k)_{L^2}|^2 \lambda_k^{-\frac 12} < \infty?$$

How do I show the sum converges? Of course I know that $\lambda_k^{-\frac 12}$ is decreasing but I don't know it explicity.