  We produce a new proof and extend results by Harrell and Stubbe for the
discrete spectrum of a self-adjoint operator. An abstract approach--based on
commutator algebra, the Rayleigh-Ritz principle, and an ``optimal'' usage of
the Cauchy-Schwarz inequality--is used to produce ``parameter-free'',
``projection-free'' versions of their theorems. We also analyze the strength of
the various inequalities that ensue. The results contain classical bounds for
the eigenvalues. Extensions of a variety of inequalities \`a la Harrell-Stubbe
are illustrated for both geometric and physical problems.
