Hadamard's Inequality for a Positive Semidefinite Matrix

Let $n$ be a natural number, let $[n]$ be the initial segment determined by $n$, let $\mathbb{R}$ be the set of real numbers with the operations and the order $\le$ of its ordered field structure, and let $B$ be a symmetric positive semidefinite real $n\times n$ matrix, with entry notation as in Real Matrix and the Set of Real Matrices. Determinants are read as in Row Properties of the Determinant, and the product below is the finite product over $[n]$.

Then

$$0\le\det B\le\prod_{i=1}^{n}B_{ii}.$$