8. Let [a, b] be a closed bounded interval, f : [a, b] -+ R be bounded, and 9 : [a, b] -+

R be continuous with g

(a) = g

(b) = O. Let fn be a uniformly bounded sequence of functions on [a, b] (see Exercise 5). Prove that if fn -+ f uniformly on all closed intervals [c, d] C

(a, b), then fng -+ fg uniformly on [a, b].