# null infinite product in the p-adic setting

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 $\mathbb C_p$.

• Am I missing something? the product can be zero in $\mathbb C$: just take $u_n=-1/n$.. – Teri Jun 30 '16 at 5:59
• You're asking whether the sum of the $\log(1+u_n)$ (which is well defined for $n$ large enough) can diverge. But $\log(1+u_n)\to 0$ so the sum of them converges $p$-adically. Did I miss something? – Gro-Tsen Jun 30 '16 at 6:40
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.