Weyl's Inequality and Monotonicity of the Ordered Eigenvalues

In the setting of Symmetric and Positive Semidefinite Matrices, the Operator Norm and Spectral Notation, let $n\in\mathbb{N}$, let $A,B\in\mathcal{S}(n)$, and let $k\in[n]$. Write $\lambda_{k}(A)$ and $\lambda_{k}(B)$ for the $k$th ordered eigenvalues of $A$ and $B$.

(i) Monotonicity. If $A\preceq B$, then $\lambda_{k}(A)\le\lambda_{k}(B)$.

(ii) Weyl's perturbation inequality.

$$\bigl|\lambda_{k}(A)-\lambda_{k}(B)\bigr|\le\lVert A-B\rVert .$$

(iii) Shift. For every $c\in\mathbb{R}$, $\lambda_{k}(A+cI_{n})=\lambda_{k}(A)+c$.