On the value of some infinite matrix games

It is shown that a zero-sum two-person noncooperative game A defined by a bounded infinite matrix in which each row converges to the same real number β and each column to the same real number α has a value V(A) if and only if α ≤ β, in which case α ≤ V(A) ≤ β. For any game defined by a bounded infinite matrix A = (aij), a necessary condition for V(A) to exist is that infj lim infi aij ≤ supi lim supj aij.