(1) Assume that $\displaystyle (f_n)$ converges uniformly to $\displaystyle f$ on $\displaystyle A$ and that each $\displaystyle f_n$ is uniformly continuous on $\displaystyle A$. Prove that $\displaystyle f$ is uniformly continuous on $\displaystyle A$

(2) Assume $\displaystyle (f_n)$ converges uniformly to $\displaystyle f$ on a compact set $\displaystyle K$, and let $\displaystyle g$ be a continuous function on $\displaystyle K$ satisfying $\displaystyle g(x) \not= 0$. Show $\displaystyle f_n/g)$ converges uniformly on $\displaystyle K$ to $\displaystyle f/g$

(3) Assume $\displaystyle (f_n)$ and $\displaystyle (g_n)$ are uniformly convergent sequences of functions.

(a) Show that $\displaystyle (f_n + g_n)$ is a uniformly convergent sequence of functions.

(b)Give an example to show that the product $\displaystyle (f_ng_n)$ may not converge uniformly.

(c) Prove that if there exists an $\displaystyle M>0$ such that $\displaystyle |f_n| \leq M$ and $\displaystyle g_n \leq M$ for all $\displaystyle n \in \mathbb{N}$, then $\displaystyle (f_ng_n)$ does converge uniformly.

I need a walkthrough with these, thanks