We consider a version of the pole assignment problem for linear discrete time-varying systems with linear state feedback. Our aim is to prove that all the systems from the closure (in the topology of pointwise convergence) of all shifts of the original system have an assignable Lyapunov spectrum if and only if the original system is uniformly completely controllable. We also provide an example to show that uniform complete controllability is not a necessary condition for assignability of the Lyapunov spectrum of the original system.