Let $Q_n$ be a sequence of polynomials of $\mathbb C_p(x)$ such that for every $z\in\mathbb C_p$ we have $\lim_{n\to+\infty}Q_n(z)=0$. One assumes that for every $n\in\mathbb N$, there exist two polynomials $P_{n,1}$ and $P_{n,2}$ of $\mathbb C_p[X]$ with distinct degree such that $Q_n(X)=P_{n,1}(X)+P_{n,2}(X)$. Can one say that for all $z\in\mathbb C_p$, $j\in\{1,2\}$ we have that $\lim_{n\to+\infty} P_{n,j}(z)=0$ ?
Thanks for any hint that you could give me.