Sequential direct limit of multiplication maps on the integers
= Sequential direct limit of multiplication maps on the integers
{title2=$\varinjlim(\mathbb Z\xrightarrow{a_0}\mathbb Z\xrightarrow{a_1}\cdots)$}
For nonzero integers $a_i$ and $P_n=a_0\cdots a_{n-1}$, the compatible embeddings of stage $n$ into $\mathbb Q$ send $m$ to $m/P_n$. They identify the direct limit with
$$
\bigcup_{n\geq0}P_n^{-1}\mathbb Z\subseteq\mathbb Q.
$$