We assume that $A$ satisfies (P2). We consider $(f_n)_{n \in \mathbb{N}}$ a sequence of elements of $A$. Let $(x_p)_{p \geqslant 0}$ be a sequence of elements of $K$. (a) Show that there exists a sequence $(\varphi_p)_{p \in \mathbb{N}}$ of strictly increasing functions from $\mathbb{N}$ to $\mathbb{N}$ such that for all $p \geqslant 0$, $f_{\psi_p(n)}(x_p)$ converges as $n$ tends to infinity with $\psi_0 = \varphi_0$ and $\psi_p = \psi_{p-1} \circ \varphi_p$ for $p \geqslant 1$. (b) Show that for all $p \geqslant 0$, $f_{\psi_n(n)}(x_p)$ converges as $n$ tends to infinity.
We assume that $A$ satisfies (P2). We consider $(f_n)_{n \in \mathbb{N}}$ a sequence of elements of $A$. Let $(x_p)_{p \geqslant 0}$ be a sequence of elements of $K$.
(a) Show that there exists a sequence $(\varphi_p)_{p \in \mathbb{N}}$ of strictly increasing functions from $\mathbb{N}$ to $\mathbb{N}$ such that for all $p \geqslant 0$, $f_{\psi_p(n)}(x_p)$ converges as $n$ tends to infinity with $\psi_0 = \varphi_0$ and $\psi_p = \psi_{p-1} \circ \varphi_p$ for $p \geqslant 1$.
(b) Show that for all $p \geqslant 0$, $f_{\psi_n(n)}(x_p)$ converges as $n$ tends to infinity.