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.



null infinite product in the p-adic setting
If $n\_0$ is such that $|u\_n|<1$ for $n \geq n\_0$, then $|1+u\_n| = 1$ for $n \geq n\_0$, so the norm of the infinite product is the norm of the product of the first $n\_0$ terms and hence nonzero.


