The author considers the discrete Hamiltonian system \[ \Delta x(t)= H_u(t,x(t+1),u(t)), \qquad \Delta u(t)= -H_x(t,x(t+1),u(t)), \tag{1} \] where \(x,u\in \mathbb R^d\), \(H(t,x,u)\) is the corresponding real Hamiltonian function having continuous derivatives in \(x\), \(u\). The author shows that the discrete nonlinear Hamiltonian system (1) is of symplectic structure. As a result its phase flow is a discrete one-parameter family of symplectic transformations preserving the phase volume. Especially, in the autonomous case, its phase flow is a discrete one-parameter group of symplectic transformations. The results are related to an open problem in \textit{C. D. Ahlbrandt} [ibid. 180, No.~2, 498--517 (1993; Zbl 0802.39005)].