Symmetric perturbation of a self-adjoint operator

We give a proof, which is valid in both real and complex Hilbert spaces, of the following result: If A is self-adjoint, B is symmetric D(B) ⊃ D(A), || Bx|| and lt; a ||x|| + ||Ax|| for x∈ D(A), then A+bB is self-adjoint for [Formula]