The time discretization by a linear backward Euler scheme for the non-stationary viscous incompressible Navier-Stokes equations with a non-zero external force in a bounded 2D domain with no-slip boundary condition or periodic boundary condition is studied. Improved global stability results are obtained. The boundedness of the solution sequence in V and D(A) norms uniform with respect to At for t E [0, ∞) is proved. A similar result in the V norm was previously obtained by (Geveci, 1989 Math. Comp., 53, 43-53) for the non-forced system. A different approach is used here. As a corollary, the global attractor for the approximation scheme is proved to exist, which is bounded in both V and D(A) spaces, thus compact in both H and V spaces. Applying the same techniques developed here, we are able to improve the main result of (Hill and Suli 2000 IMA J. Numer. Anal., 20, 633-667) by showing that besides the existence of a global attractor, the whole solution sequence is uniformly bounded in V as well, which is of significance from the point of view of computing. As a corollary of local convergence results, upper semi-continuity of the attractor with respect to the numerical perturbation induced by the linear scheme is also established in both H and V spaces. Finally, some preliminary estimates, which are to our knowledge the first of their kind, on the dimensions of the attractors in H and V spaces are also obtained.