Lamperti in 1964 showed that the convergence of the marginals of an extremal process generated by independent and identically distributed random variables implies the full weak convergence in the Skorohod $J_1$-topology. This result is generalized to the $k$th extremal process and to random variables which need not be identically distributed. The proof here is based on the weak convergence of a certain point-process (which counts the number of up-crossings of the variables) to a two-dimensional nonhomogeneous Poisson process.
"On Weak Convergence of Extremal Processes." Ann. Probab. 4 (3) 470 - 473, June, 1976. https://doi.org/10.1214/aop/1176996096