I am studying for the proof of Weyl inequlity from a course note: 1.3 Implications of the Courant-Fischer Theorem. It says that we can prove (d) from (c), that is we can prove: $\lambda_{j+k-1}(A+B)\leq\lambda_j(A)+\lambda_k(B)$ by using: $\lambda_k(A+B)\leq\lambda_{k-r}(A)$. I don't know how to prove it.
I have searched for some materials which may be useful, such as Corollary 2.6 from this paper. However, it involves "positive inertia index" which I don't know and can not understand the proof.
I know Proof or reference to the Weyl inequalities? here has provided a blog link to one proof. But the course note has provided a different ways of proof and I really want to solve it out. Thank you.
