Dr. Tao in his notes on eigenvalue inequalities uses Courant-Fischer min-max theorem to prove the eigenvalue stability inequality. Specifically, I am looking for proof of Eq. (13) where Dr. Tao states as an immediate result of Eq. (6) and (10). But the problem is that the min-max function is not convex. I have read Stewart & Sun's book on Matrix Perturbation Theory, but it seems that they have felt that it is obvious too.
Can someone provide more details on how to derive Eq. (13)?