Let $M \in \mathcal{S}_{n}(\mathbb{R})$, we denote by $m = s^{\downarrow}(M)$ its ordered spectrum, and $\left(v_{1}, \ldots, v_{n}\right)$ an orthonormal basis from the spectral resolution of $M$. Let $j$ be an integer, $1 \leqslant j \leqslant n$. We denote by $\mathcal{V}_{j}$ the vector subspace of $\mathbb{R}^{n}$ spanned by $\left(v_{1}, \ldots, v_{j}\right)$, and by $\mathcal{W}_{j}$ the one spanned by $\left(v_{j}, v_{j+1}, \ldots, v_{n}\right)$. Show the equalities $$\inf_{x \in \mathcal{V}_{j},\, \|x\|=1} \langle x, Mx \rangle = \sup_{x \in \mathcal{W}_{j},\, \|x\|=1} \langle x, Mx \rangle = m_{j}.$$
Let $M \in \mathcal{S}_{n}(\mathbb{R})$, we denote by $m = s^{\downarrow}(M)$ its ordered spectrum, and $\left(v_{1}, \ldots, v_{n}\right)$ an orthonormal basis from the spectral resolution of $M$. Let $j$ be an integer, $1 \leqslant j \leqslant n$. We denote by $\mathcal{V}_{j}$ the vector subspace of $\mathbb{R}^{n}$ spanned by $\left(v_{1}, \ldots, v_{j}\right)$, and by $\mathcal{W}_{j}$ the one spanned by $\left(v_{j}, v_{j+1}, \ldots, v_{n}\right)$. Show the equalities
$$\inf_{x \in \mathcal{V}_{j},\, \|x\|=1} \langle x, Mx \rangle = \sup_{x \in \mathcal{W}_{j},\, \|x\|=1} \langle x, Mx \rangle = m_{j}.$$