I'm going through the book A First Look at Rigorous Probability Theory (Second Edition) by Rosenthal. On page 131 he makes this argument, which I can't follow.
Theorem 10.1.1 states equivalent definitions of weak convergence. Theorem 11.1.10 says this:
I can't figure out how he arrives at the contradiction since, in my mind, the fact that $\{\mu_{n_{k_j}}\}$ converges weakly to $\mu=\nu$ doesn't imply that the sequence $\{\mu_{n_k}\}$ does the same. Would someone care to explain? Thanks