I cannot follow an argument for positive-semidefiniteness of the one-electron density matrix given in "Molecular Electronic-Structure Theory" by Helgaker/Jorgensen/Olsen. First some definitions: $F(M)$ denotes the Fock Space spanned by a basis of $M$ orthonormal spin orbitals (which are not further specified). $F(M,N)$ denotes its subspace of $N$-electron states. Note that because we are considering a finite one-electron basis, the Fock-Space only includes states with a maximum number of $N$ electrons. The authors further define the one-electron density matrix for an arbitrary $N$-electron state $|x\rangle$ by $\bar D_{PQ} = \langle x| a_P^\dagger a_Q |x\rangle$. Now comes the sentence I don't understand: "The one-electron density matrix is positive semidefinite since its elements are either trivially equal to zero or inner products of states in the subspace $F(M, N-1)$" (given in section 1.7.1).
I'm assuming that the authors mean that the matrix elements are inner products whenever $N\geq 1$ and are equal to zero when $N=0$ (vacuum state). I also know that a matrix is positive-semidefinite when it is the matrix representation of a positive-semidefinite bilinear form, but I don't see a connection to the above argument, especially since the inner product is positive definite (without "semi").
Please do not provide any alternate arguments why the density matrix is positive-semidefinite (I think I can find these myself). I would like to understand the specific argument given above.