How to show that for any graph $G$ $\lambda_{n}(G) \ge d_{\max}(G)$

Question

To my understanding, the maximum degree of a node in a $n$ vertices graph is $n-1$. I am assuming that we have to somehow prove that maximum eigenvalues are greater than $n - 1$. The hint given was that we should use the Courant Fischer Theorem, but I am still struggling with approaching the problem. How do I prove this?