We assume that $A$ satisfies (P2). We consider $(f_n)_{n \in \mathbb{N}}$ a sequence of elements of $A$. (a) Show that we can extract from the sequence $(f_n)_{n \in \mathbb{N}}$ a subsequence that converges pointwise on $\mathbb{Q} \cap K$. We denote $(g_n)_{n \in \mathbb{N}}$ this extraction. (b) For $x \in K$, show that $(g_n(x))_{n \in \mathbb{N}}$ admits a unique cluster value denoted $g(x)$ and conclude on the pointwise convergence of the sequence $(g_n)_{n \in \mathbb{N}}$ on $K$ to $g$.
We assume that $A$ satisfies (P2). We consider $(f_n)_{n \in \mathbb{N}}$ a sequence of elements of $A$.
(a) Show that we can extract from the sequence $(f_n)_{n \in \mathbb{N}}$ a subsequence that converges pointwise on $\mathbb{Q} \cap K$. We denote $(g_n)_{n \in \mathbb{N}}$ this extraction.
(b) For $x \in K$, show that $(g_n(x))_{n \in \mathbb{N}}$ admits a unique cluster value denoted $g(x)$ and conclude on the pointwise convergence of the sequence $(g_n)_{n \in \mathbb{N}}$ on $K$ to $g$.