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)?

T. 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 he 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)?