Let $f$ be a symmetric endomorphism of $\mathbf{R}^n$ with eigenvalues $\lambda_1 \leqslant \ldots \leqslant \lambda_n$ and associated orthonormal eigenbasis $(e_1, \ldots, e_n)$. Let $k \in \llbracket 1, n \rrbracket$, $S_k$ a vector subspace of $\mathbf{R}^n$ of dimension $k$, and $T_k = \operatorname{Vect}(e_k, \ldots, e_n)$.
By considering $x \in S_k \cap T_k$, justify that:
$$\max_{x \in S_k, \|x\|=1} (x, f(x)) \geq \lambda_k.$$