By using variational means, it is found for any one velocity system with non- zero absorption cross section having either vacuum, reflecting, or antireflecting boundary conditions that the transport solution is, in a very specific sense, approached monotonically from above by the solutions to the odd PN equations and from below by the solutions to the even PN equations, provided the PN solutions are obtained by using appropriate continuity and external boundary conditions. That is to say, odd and even PN calculations “bracket” the transport solution. In one instance, the escape probability is bounded and, in another, the disadvantage factor. This theoretical result, along with certain numerical evidence, suggests that the modified P2 approximation of Dawson may serve as a practical, reasonably accurate alternative to diffusion theory for certain realistic design problems.