  We prove strong convergence results for the motivic Adams spectral sequence
of the sphere spectrum over fields with finite virtual cohomological dimension
at the prime 2, and over arbitrary fields at odd primes. We show that the
motivic Adams spectral sequence is not strongly convergent over number fields.
As applications we give bounds on the exponents of the $(\ell,\eta)$-completed
motivic stable stems, and calculate the zeroth $(\ell,\eta)$-completed motivic
stable stems.
