 Solution

A direct system of abelian groups over the directed poset $A$ consists of groups $G_a$ and maps ${\varphi }_{ab}:G_a\to G_b$ for $a\le b$, satisfying ${\varphi }_{aa}=1$ and ${\varphi }_{bc}{\varphi }_{ab}={\varphi }_{ac}$. Its direct limit is the quotient of ${⨁}_a{G}_a$ by the relations $g_a\sim {\varphi }_{ab}(g_a)$.

For the displayed sequence, put $P_0=1$ and $P_n=a_0{a}_1\cdots a_{n-1}$. Map the copy of $\mathbb{Z}$ at stage $n$ to $\mathbb{Q}$ byThis is compatible with the next transition because $a_{n}m/P_{n+1}=m/P_n$. The universal property of the direct limit therefore identifies it withIn reduced form, these are exactly the rationals whose denominator divides one of the finite products $P_n$. This is the sequential direct limit of multiplication maps on the integers.