We assume that $A$ satisfies (P2). We consider $(f_n)_{n \in \mathbb{N}}$ a sequence of elements of $A$, and $(g_n)_{n \in \mathbb{N}}$ a subsequence converging pointwise on $K$ to $g$.
(a) Show that $g$ is continuous on $K$.
(b) Show that the sequence $(g_n)_{n \in \mathbb{N}}$ converges uniformly to $g$ on $K$. (Hint: you may reason by contradiction.)
(c) Deduce that $(P2) \Rightarrow (P1)$.