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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.

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    $\begingroup$ Isn't $Q_n(X) = p^n X, P_{n,1}(X) = p^n X + 1, P_{n,2}(X) = -1$ a counterexample? $\endgroup$ Jan 10, 2014 at 10:36
  • $\begingroup$ I'd say it's not so much a counterexample as a proof that the answer to the question is no. But then that's probably what you meant. :) $\endgroup$
    – R.P.
    Jan 10, 2014 at 23:06
  • $\begingroup$ Sure, but I thought the OP might want to add some extra condition ruling out examples like these. If not, of course I can transform it into an answer. $\endgroup$ Jan 10, 2014 at 23:25

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