Assume that we define the ring of the $p$-adic integers as the projective limit $$\mathbb{Z}_p =\varprojlim \frac{\mathbb{Z}}{p^n\mathbb{Z}}$$
Then $\mathbb{Q}_p$, the field of the $p$-adic numbers is no more than the field of quotients of $\mathbb{Z}_p$.
But, to do so, we must prove that $\mathbb{Z}_p$ is an Integral Domain (with this definition).
In order to do so, I take two sequences $(x_n)_{n\in\mathbb{N}}$ and $(y_n)_{n\in\mathbb{N}}$ in the projective limit which are nonzero (assume that $x_{n_{0}}\not =0 \pmod{p^{n_0}}$ and $y_{m_{0}}\not =0 \pmod{p^{m_0}}$ and I have to prove that the product is nonzero. I am trying to prove that $x_{n_{0}+m_{0}}y_{n_{0}+m_{0}}\not=0 \pmod{p^{n_{0}+m_{0}}}$.
Since both sequences are in the projective limit, I can easily see that $x_{n_{0}+m_{0}}\not=0 \pmod{p^{n_{0}+m_{0}}}$ and $y_{n_{0}+m_{0}}\not=0 \pmod{p^{n_{0}+m_{0}}}$. But how can I be sure that the product will also be nonzero?
Thanks in advance.