## Two questions about function convergence

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.