$X,Y$ are $n\times n$ Hermitian matrices. $\lambda_k(X)$ denotes the $k$th largest eigenvalue of $X$. Prove that $\sum_{k=1}^n|\lambda_k(X)-\lambda_k(Y)|\leq\|X-Y\|_1$. (Here $\|X-Y\|_1=\sum_{k=1}^n|\lambda_k(X-Y)|$)
Attempts: Using Spectral decomposition we can write $X-Y$ as $P-Q$, where $P,Q\succeq 0$, and $\|X-Y\|_1=tr(P)+tr(Q)$, but I don't know how to associate with $\lambda_k(X)-\lambda_k(Y)$.