Simple proof that density operator has only pure point spectrum

Question

Is there any simple proof that self-adjoint, nonnegative and trace one operators have only point spectrum? Namely, we want to show that if a bounded operator $\rho$ on $\mathcal{H}$ is such that $$ \rho=\rho^\dagger, $$ $$ (v,\rho v)\geq0, \text{for any } v\in\mathcal{H}, $$ $$ {\rm Tr}(\rho)=1, $$ the it can be decomposed as $\rho=\sum_k\rho_k P_k$ where $\rho_k$ are the eigenvalues of $\rho$ and $P_k$ the projectors onto the subspace spanned by the eigenvectors associated to $\rho_k$.