Let G be the set of integer sequences $(a_i)_{i=1}^∞$ for which there is some $N ∈ ℕ$ such that $a_i$ = 0 for all i > N . Let G be the set of integer sequences $(a_i)_{i=1}^∞$ for which there is some $N ∈ ℕ$ such that $a_i$ = 0 for all i > N .
(a) Show that G is an abelian group under the operation:
$(a_i)_{i=1}^∞$ + $(b_i)_{i=1}^∞$ = $(a_i + b_i)_{i=1}^∞$
(b) Show that G is isomorphic to $ℚ^×$

For part a, isn't this trivial since an integer sequence would just be a sum and addition is commutative? 
And for part b, how can I show it's isomorphic to the complex number closed under multiplication when the law of composition for G is an addition function? I think I understand intuitively that the sum ai + bi would map one-to-one to Q but I'm not sure how to show it.
I have a feeling I'm not interpreting this correctly at all.

Note that ${\mathbb Q}^\times$ has an element of order $2$ (namely $-1$), while
$G$ is torsion-free. So $G$ is not isomorphic to ${\mathbb Q}^\times$.
On the other hand, it is isomorphic to ${{\mathbb Q}_+}^\times$. If we denote
the $i$-th prime number by $p_i$, the map $f: G \to {{\mathbb Q}_+}^\times$
defined by
$$
f((a_i)_{i\geq 1})=\prod_{i} p_i^{a_i}
$$
is an isomorphism.
