Let $M$ be a compact manifold and $g_k$ be a sequence of Riemannain metrics smoothly converging to another Riemannian metric $g$.

Let $\{\lambda^k_j\}$ be the spectrum of the Laplacian of the Riemannian metric $g_k$, and $\{\lambda_j\}$ be the spectrum of the Laplacian of $g$ (with values in increasing order, i.e., $\lambda^k_{j+1} \geq \lambda^k_{j}$).

I expect that $\lambda^k_j \to \lambda_j$ as $k \to \infty$. Is this true? If yes, what is a reference? If I replace smooth convergence with $C^k$-convergence, what is the lowest $k$ such that convergence of spectrum holds?