Let [a, b] be a closed bounded interval, f : [a, b] → R be bounded, and g : [a, b] → R be continuous with g(a) = g(b) = 0. Let fn be a uniformly bounded sequence of functions on [a, b] (see Exercise 7.1.3). Prove that if fn → f uniformly on all closed intervals [c, d] ⊂ (a, b), then fng → fg uniformly on [a, b].