I'm looking for a fleshed out proof of the following theorem.
Theorem: Let $G=(V,\mathbf{W})$ be an undirected, edge-weighted graph with normalized Laplacian $\mathbf{L}_N$. Furthermore, let $0=\lambda_1\leq\lambda_2\leq\cdots\leq\lambda_{n-1}\leq 2$ be the eigenvalues of $\mathbf{L}_N$. Choosing some $k$ such that $k
$ \text{Ncut}(A_1,...,A_k) \geq \sum\limits_{i=1}^{k-1}\lambda_i$
where $\text{Ncut}(A_1,...,A_k) := \sum\limits_{i=1}^k\frac{W(A_i,\overline{A_i})}{\text{vol}(A_i)}$. Any help is appreciated, thanks!