  We consider Voevodsky's slice tower for a finite spectrum E in the motivic
stable homotopy category over a perfect field k. In case k has finite
cohomological dimension (in characteristic two, we also require that k is
infinite), we show that the slice tower converges, in that the induced
filtration on the bi-graded homotopy sheaves for each term in the tower for E
is finite, exhaustive and separated at each stalk. This partially verifies a
conjecture of Voevodsky.
