I wonder if someone can prove/disprove the following inequality,

$\lambda_i(A+mI) \leq \lambda_i(A+K) \leq \lambda_i(A+MI)$

where $A$ is a real symmetric Metzler matrix with real and negative eigenvalues and $K$ is a diagonal matrix. $M$ and $m$ are the greatest and the least elements of $K$ respectively.

I have seen that the above inequality holds for all examples I tried but I would like to prove it mathematically or find a counterexample.

Any help would be appreciated. Thanks.