Consider the group $H$ of all sequences of integers $(a_1, a_2, ...)$ such that only finitely many $a_i$ are different from $0$ (and $(0, 0, 0, . . .) \in H$) under $(a_1, a_2, . . .) + (b_1, b_2, . . .) = (a_1 + b_1, a_2 + b_2, . . .)$. Show that $H$ is isomorphic to the multiplicative group of positive rational numbers.
At first I thought of using the idea that we can define a fraction $a/b$ as an ordered pair of integers $(a,b)$, but I have not been able to fit it with the given conditions. Any help I would appreciate