  For vanishing fidelity between initial and final states two important quantum
speed limits, the Mandelstam-Tamm limit (involving energy dispersion) and
Margolus-Levitin one (involving excitation energy expectation value) have been
derived. While the generalization of the former limit to the case of arbitrary
fidelity is straightforward, the relevant generalization of the latter, given
in the seminal paper by Giovanetti et al (Phys. Rev. A67 (2003), 052109) was
based on the conjectured equality of lower and upper bounds on the right hand
side of generalized Margolus-Levitin inequality, verified numerically up to
seven digits. Only recently there appear two proofs of the conjecture. We
provide below a very elementary new proof, based on the simplest tools from
differential calculus. Thus the generalized Margolus-Levitin speed limit can be
derived much in the spirit of the original one valid for vanishing fidelity.
