Let $p$ be a prime, $\mathbb C_p$ be the completion of a algebraic closure of $\mathbb Q_p$ and $(u_n)_{n\in\mathbb N}$ be a sequence of $\mathbb C_p$ converging towards $0$. Suppose that for all $n\in\mathbb N$, one has $1+u_n\ne0$. Can $\prod_{n\in\mathbb N}(1+u_n)$ be zero?

I know that is trivially true in $\mathbb C$, but I do not have any exemple (or counterexemple) in $\mathbb C_p$.

Thanks in advance.

canbe zero in $\mathbb C$: just take $u_n=-1/n$.. $\endgroup$ – Teri Jun 30 '16 at 5:59