Abstract In this note, we show that if a sequence of moment generating functions M n ( t ) converges pointwise to a moment generating function M ( t ) for all t in some open interval of R, not necessarily containing the origin, then the distribution functions F n (corresponding to M n ) converge weakly to the distribution function F (corresponding to M). The proof uses the basic classical result of Curtiss [1942. A note on the theory of moment generating functions. Ann. Math. Statist. 13 (4), 430–433].