An improved result is presented, showing that if the finite-dimensional distributions of a sequence of martingales converge, and if for each time $t$ the variables are uniformly integrable, then weak convergence follows (in either $C$ or $D$) provided the limiting process satisfies a certain condition; this condition is satisfied by the Wiener process.
"A Criterion for Tightness for a Sequence of Martingales." Ann. Probab. 4 (5) 859 - 862, October, 1976. https://doi.org/10.1214/aop/1176995990