This follows through use of the Cauchy–Schwarz inequality, since |⟨ A ⟩| |⟨ B ⟩| ≥ |⟨ A B ⟩|, and A B = ([ A, B ] + { A, B })/2 ; and similarly for the shifted operators A − ⟨ A ⟩ and B − ⟨ B ⟩. (Cf. uncertainty principle derivations.)