  We prove the convergence of an incremental projection numerical scheme for
the time-dependent incompressible Navier--Stokes equations, without any
regularity assumption on the weak solution. The velocity and the pressure are
discretised in conforming spaces, whose the compatibility is ensured by the
existence of an interpolator for regular functions which preserves approximate
divergence free properties. Owing to a priori estimates, we get the existence
and uniqueness of the discrete approximation. Compactness properties are then
proved, relying on a Lions-like lemma for time translate estimates. It is then
possible to show the convergence of the approximate solution to a weak solution
of the problem. The construction of the interpolator is detailed in the case of
the lowest degree Taylor-Hood finite element.
