Parity effect in a small superconducting grain: A rigorous result
  The parity effect in an ultra-small superconducting grain is examined. By applying a generalized version of Lieb's spin-reflection positivity technique, we show rigorously that the parity parameter $\Delta_P$ is nonvanishing in such a system. A positive lower bound for $\Delta_P$ is derived. 
Guang-Shan Tian and Lei-Han Tang