I need help proving the CMT for random vectors. I'm currently reading Econometric Analysis for Cross Section and Panel Data by Jeffrey M. Wooldridge, and, unfortunately, he leaves it to the reader to prove most of the asymptotic results.
Q. Prove that if x_n converges in distribution to x, then for a continuous function g: R^k to R^l, g(x_n) converges in distribution to g(x). I need a rigorous, epsilon-delta proof of this. Any good, rigorous links to a proof would be appreciated.
Thanks. Christian S.