Theorem (Helly-Bray): $\x_n \inD \x$ if and only if $\Ex g(\x_n) \to \Ex g(\x)$ for all continuous bounded functions $g: \real^d \to \real$.
Traditionally, “Helly-Bray Theorem” refers only to the forward part of the theorem.
Proof: Ferguson, A Course in Large Sample Theory (1996), Theorem 3.
See also: Portmanteau theorem, which generalizes the Helly-Bray Theorem.