We give response to the question: in infinite dimension states, given a state with energy bounded by E, can we write the state as a countable convex combination of pure states with energy bounded by E?. We review the Alicki- Fannes-Winter technique to obtain a uniform continuity bound for the von Neumann entropy in states that are a mix of pure states with bounded energy, using this bound we conclude that for a Hamiltonian satisfying the Gibb’s hypothesis such states cannot exist.