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

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

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