  We provide a new proof of convergence to motion by mean curvature (MMC) for
the Merriman-Bence-Osher (MBO) thresholding algorithm. The proof is elementary
and does not rely on maximum principle for the scheme. The strategy is to
construct a natural ansatz of the solution and then estimate the error. The
proof thus also provides a convergence rate. Only some weak integrability
assumptions of the heat kernel, but not its positivity, is used. Currently the
result is proved in the case when smooth and classical solution of MMC exists.
