I would like to show $\mathbb{Z}_p/p^n\mathbb{Z}_p \approx \mathbb{Z}/p^n\mathbb{Z}$.
I have this proof but i dont know how can i finish it:
I know $\forall x\in\mathbb{Z}_p$ $\exists !(\alpha_n)$ Cauchy sequence converging to $x$ such as:
$\cdot$ $\alpha_n\in\mathbb{Z}:0\leq\alpha_n\leq p^{n}-1$.
$\cdot$ $\alpha_n\equiv \alpha_{n-1}(mod\hspace{0.1cm}p^n)$ para todo $n=1,2,...$
I define $\psi_n:\mathbb{Z}_p\longrightarrow \frac{\mathbb{Z}}{p^n\mathbb{Z}}:\psi(x)= \alpha_n\hspace{0.2cm} mod \hspace{0.1cm}p^n$
It is easy to check $ker(\psi_n)=p^n\mathbb{Z}_p$ and sobrejective. My problem is to show $\psi_n$ is a homomorphism of ring.
For that: Let be $x,y\in\mathbb{Z}_p$ and $(\alpha_n)$, $(\beta_n)$ associated with $x$ and $y$. I showed $\alpha_n +\beta_n \longrightarrow x+y$ and $\alpha_n\beta_n \longrightarrow xy$ and $\alpha_n+\beta_n\equiv \alpha_{n-1}+\beta_{n-1}(mod\hspace{0.1cm}p^n)$, $\alpha_n\beta_n\equiv \alpha_{n-1}\beta_{n-1}(mod\hspace{0.1cm}p^n)$ $\forall n=1,2,...$
But i couldnt show $0\leq\alpha_n\beta_n\leq p^{n}-1$, and $0\leq\alpha_n+ \beta_n\leq p^{n}-1$.
So i cant say $\alpha_n+\beta_n$ is the correspondence Cauchy sequence for x+y or $\alpha_n\beta_n$ is the correspondence Cauchy sequence for $xy$.