Let $\Omega\subset\mathbb{R}^n$ be a bounded domain with smooth boundary, consider the following Dirichlet eigenvalue problem. \begin{equation}\label{1} \left\{ \begin{array}{ll} -\Delta u=\lambda u, & \hbox{in $\Omega$;} \\[3mm] u=0, & \hbox{on $\partial\Omega$.} \end{array} \right. \end{equation} The Weyl's law shows that $$ N(\lambda)=\#\{k\mid 0<\lambda_k \leq \lambda\}\sim \frac{\omega_n}{(2\pi)^n} |\Omega| \lambda^{\frac{n}{2}},\qquad \lambda\to +\infty. $$ and it means that $$ \lambda_k\sim \frac{(2\pi)^2}{(\omega_n|\Omega|)^{\frac{2}{n}}} k^{\frac{2}{n}},\qquad k\to +\infty.$$
I feel confused about how can we prove the second conclusion from the first conclusion. It seems we only know that $N(\lambda_k)\geq k$. I found in some books that use the fact $N(\lambda_k)=k$, I don't know how to get this since $\lambda_{k}$ may has multiplicty. For example, if $0<\lambda_{1}<\lambda_2=\lambda_3=\lambda_4<\lambda_5<\cdots$, then we have $N(\lambda_2)=N(\lambda_4)=4$. Can some one help me with this question? Thanks a lot!