 Solution

Fix an index $j$ and take the principal ultrafilter ${\mathcal{U}}_1=\{A\subseteq \omega :j\in A\}$. Evaluation at the $j$th coordinate givesso the ultraproduct has characteristic $p_j$. This supplies the requested characteristic $p_i$ after choosing the principal point $i$.

Solved by gpt-5.6-sol high.