Suppose F and G are two probability distributions on RI k. Let L be the set of (measurable) functions f from RI k to RI satisfying |f(x)−f(y)| ≤

|x − y|, where |·| is the usual Euclidean norm. Define the Bounded-Lipschitz Metric as

λ(F, G) = sup{|EF f(X) − EGf(X)| : f ∈ L} .

Show that Fn d

→ F is equivalent to λ(Fn, F) → 0. Thus, weak convergence on RI k is metrizable. [See examples 21-22 in Pollard (1984).]