Once again ignoring complications involved with a continuous spectrum of H , suppose the spectrum of H is bounded from below and that its greatest lower bound is E 0. The expectation value of H in a state | ψ ψ ⟩ ⟩ is then ⟨ ψ ψ | H | ψ ψ ⟩ = ∑ ∑ λ λ 1 , λ λ 2 ∈ ∈ S p e c ( H ) ⟨ ψ ψ | ψ ψ λ λ 1 ⟩ ⟨ ψ ψ λ λ 1 | H | ψ ψ λ λ 2 ⟩ ⟨ ψ ψ λ λ 2 | ψ ψ ⟩ = ∑ ∑ λ λ ∈ ∈ S p e c ( H ) λ λ | ⟨ ψ ψ λ λ | ψ ψ ⟩ | 2 ≥ ≥ ∑ ∑ λ λ ∈ ∈ S p e c ( H ) E 0 | ⟨ ψ ψ λ λ | ψ ψ ⟩ | 2 = E 0 ⟨ ⟨ ψ ψ | ψ ψ ⟩ ⟩ .