  As an alternative to the well-known methods of "chaining" and "bracketing"
that have been developed in the study of random fields, a new method, which is
based on a {\em stochastic maximal inequality} derived by using the formula for
integration by parts and on a new concept named {\em strict countability}, is
presented. The main results are some weakconvergence theorems for sequences of
separable random fields of discrete-time martingales under the uniform topology
with the help also of entropy methods. As special cases, some new results for
i.i.d.\ random sequences, including a new Donsker theorem and a moment bound
for suprema of empirical processes indexed by classes of sets or functions, are
obtained.
