Let $f$ be a symmetric endomorphism of $\mathbf{R}^n$. We denote by $\lambda_1 \leqslant \ldots \leqslant \lambda_n$ the eigenvalues ordered in increasing order of $f$.
Justify the existence of an orthonormal basis $(e_1, \ldots, e_n)$ of $\mathbf{R}^n$ formed of eigenvectors of $f$, the vector $e_i$ being associated with $\lambda_i$ for all $i \in \{1, \ldots, n\}$. We keep this basis henceforth.