T. Tao in [his notes on eigenvalue inequalities uses Courant-Fischer min-max theorem to prove the eigenvalue stability inequality](https://terrytao.wordpress.com/2010/01/12/254a-notes-3a-eigenvalues-and-sums-of-hermitian-matrices/). 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)?



Proof of eigenvalue stability inequality via Courant-Fischer min-max theorem
It is a simple and repeated application of $\min$ and $\max$ operators.


$$v^\*(A+B)v=v^\*Av+v^\*Bv\le v^\*Av+\|B\|\_{op},\,\forall v\in R^n\wedge |v|=1.$$
Given $V$ where $\dim(V)=i$,
$$\min\_{u\in V,|u|=1}u^\*(A+B)u\le v^\*(A+B)v\le v^\*Av+\|B\|\_{op},\,\forall v\in V\wedge |v|=1,$$
then
$$\min\_{u\in V,|u|=1}u^\*(A+B)u\le \min\_{v\in V,|v|=1}v^\*Av+\|B\|\_{op}\le \max\_{\dim(V)=i}\min\_{v\in V,|v|=1}v^\*Av+\|B\|\_{op}$$
and
$$\max\_{\dim(U)=i}\min\_{u\in V,|u|=1}u^\*(A+B)u\le \max\_{\dim(V)=i}\min\_{v\in V,|v|=1}v^\*Av+\|B\|\_{op}.$$
In other words
$$\lambda\_i(A+B)\le\lambda\_i(A)+\|B\|\_{op}.$$
Similarly, we can prove
$\lambda\_i(A)-\|B\|\_{op}\le\lambda\_i(A+B)$
and reach the desired inequality.


