1. Let f_n : [a,b] ->R be functions satisfying the Lipschitz condition for matching K_n.
(a) Assume that f_n -> f pointwise. Show that, if liminf K_n < \infty , f satisfies the Lipschitz condition for K \le liminf K_n.

(b) Assuming that all K_n are equal, show that the convergence f_n -> f is uniform.

2. Let f_n -> f, g_n -> g uniformly in given S.
Show that if there exists M>0, such as  M \ge |f_n (x)| , M \ge |g_n (x)| for all x \in S, then f_n g_n -> fg uniformly in S.