Convergence and p-adic numbers

Question

Let $p_n$ be the $n$-th prime number, and $p_{\infty}=\infty$. Let $(a_n)$ a family such that $a_n \in Z_{p_n}$, with $a_\infty=0$. construct a sequence of rationals numbers $(z_n)$ such that for every i, $lim_{n \to \infty} |z_n-a_i|_{p_i}=0$ .

any help ?

Answer 1

Let $b_{i,n} = order( ( \prod_{j\le n, j\ne i} p_j^n) \bmod p_i^n)$ then consider $$w_n=\sum_{i\le n} (a_i\bmod p_i^n) (\prod_{j\le n, j\ne i} p_j^n)^{b_{i,n}}$$

How would you repair $w_n$ to get that $|w_n|_\infty \to 0$ ?