Convergence in distribution

A seq. X₁, X₂, ... of R-valued random vars. converges in distribution (or converges weakly, or in law) to a random var. X

⇔ lim Fₙ(x) = F(x) as n→∞ for all continuity points x of F

⇔ E(f(Xₙ))→E(f(X)) for all bounded Lipschitz f

⇔ liminf E(f(Xₙ))≥E(f(X)) for all non-negative continuous f

⇔ liminf P(Xₙ∈G)≥P(X∈G) for every open set G

⇔ limsup P(Xₙ∈F)≤P(X∈F) for every closed set F

⇔ P(Xₙ∈B)→P(X∈B) for all continuity sets B of X (i.e., P(X∈∂B)=0)

How many can you prove?