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

and supx | f (x)| ≤ 1, where |·| is the usual Euclidean norm. Define the BoundedLipschitz Metric as λ(F, G) = sup{|EF f (X) − EG f (X)| : f ∈ L} .
Show that Fn d → F is equivalent to λ(Fn, F) → 0. Thus, weak convergence on IRk is metrizable. [See examples 21–22 in Pollard (1984).]