As in the third part, we suppose that $B = A + \mathbf{u}\mathbf{u}^T$ with $A \in \mathcal{S}_n(\mathbb{R})$ a symmetric matrix, and $\mathbf{u} \in \mathbb{R}^n$ a vector such that $\|\mathbf{u}\| = 1$. We denote by $\lambda_1 \leqslant \lambda_2 \leqslant \cdots \leqslant \lambda_n$ the eigenvalues of $A$ and $\mu_1 \leqslant \mu_2 \leqslant \cdots \leqslant \mu_n$ those of $B$. We admit that $$\lambda_1 \leqslant \mu_1 \leqslant \lambda_2 \leqslant \mu_2 \leqslant \cdots \leqslant \lambda_n \leqslant \mu_n.$$ We further suppose that there exists an integer $m \in \{1, 2, \ldots, n-1\}$ such that the eigenvalues of $A$ satisfy $$0 = \lambda_1 = \lambda_2 = \cdots = \lambda_m < \lambda_{m+1} \leqslant \cdots \leqslant \lambda_n.$$ Let $\varepsilon \in ]0, \lambda_{m+1}[$. We suppose that $\left\langle \mathbf{u}, \left(A - \varepsilon \mathbb{I}_n\right)^{-1}\mathbf{u}\right\rangle < -1$.
Show that $\operatorname{Tr}\left(\left(B - \varepsilon \mathbb{I}_n\right)^{-1}\right) > \operatorname{Tr}\left(\left(A - \varepsilon \mathbb{I}_n\right)^{-1}\right)$.