4. Let

f, 9 be continuous on a closed bounded interval [a, b] with Ig(x)1 > 0 for x E [a, b]. Suppose that fn -+ f and gn -+ gas n -+ 00, uniformly on [a, b].

(a) Prove that II gn is defined for large nand fnl gn -+ fig uniformly on [a, b]

as n -+ 00.

(b) Show that

(a) is false if [a, b] is replaced by

(a, b).