  We use the Dirac equation in a fixed black hole background and different
independent techniques to demonstrate the absence of fermionic bound states
around a Schwarzschild black hole. In particular, we show that no embedded
eigenvalues exist which have been claimed for the case when the energy is less
than the particle's mass. We explicitly prove that the claims regarding the
embedded eigenvalues can be traced back to an oversimplified approximation in
the calculation. We conclude that no bound states exist regardless the value of
the mass.
