  The subdivision algorithm by Dellnitz and Hohmann for the computation of
invariant sets of dynamical systems decomposes the relevant region of the state
space into boxes and analyzes the induced box dynamics. Its convergence is
proved in an idealized setting, assuming that the exact time evolution of these
boxes can be computed.
  In the present article, we show that slightly modified, directly
implementable versions of the original algorithm are convergent under very mild
assumptions on the dynamical system. In particular, we demonastrate that
neither a fine net of sample points nor very accurate approximations of the
precise dynamics are necessary to guarantee convergence of the overall scheme.
