I have a bounded domain $\Omega\subset\mathbb R^n$. I also have a sequence of subdomains $\Omega_n$, such that $\Omega_n\subset\Omega_{n+1}$ for all $n$, and moreover $\Omega=\bigcup_n \Omega_n$. Denoting the $k$th eigenvalue of the Dirichlet Laplacian on the domain $A$ by $\lambda_k(A)$, I want to prove that $\lim_{n\rightarrow \infty}\lambda_k(\Omega_n)=\lambda_k(\Omega)$.
This seems like a very straightforward application of the min-max principle, but I am having some problems. From domain monotonicity, we naturally have that $\lim_{n\rightarrow \infty}\lambda_k(\Omega_n)\geq\lambda_k(\Omega)$, so one only needs to prove the other inequality. I was able to prove this inequality for $\lambda_1$, but I am having trouble generalizing to higher eigenvalues.
My idea was to consider the restriction of the eigenfuntion $f_1$ to $\Omega_n$, and then extending this restriction 'nicely', so that the support of the extension is contained in $\Omega_{n+1}$ (and it is contained in $H_{0}^{1}(\Omega_{n+1})$). This can be done by using an appropriate bump function. One can then show that the Rayleigh quotient of this approximation, $f_{1,n}$, converges to the Rayleigh quotient of $f_1$. Since $\lambda_1(\Omega)$ is the unique minimizer of the Rayleigh quotient, the result follows for $k=1$.
Now I am stuck about the higher eigenvalues. I know that $\lambda_k(\Omega)$ minimizes the Rayleigh quotient, when restricted to the subspace orthogonal to the first $k-1$ eigenfunctions. So naively, I would like to proceed by induction, repeating the proof above while restricting myself to this orthogonal complement. But, I am not sure how I can guarantee that my approximating sequence $f_{k,n}$ is contained in this orthogonal subspace.
Is there a simple way to guarantee this, without trying to construct this sequence explicitly? (Using concrete bump functions etc., which does not sound very nice.) Is there maybe another simple way to prove this?
Thanks in advance!