homomorphisms between free abelian groups that are not finitely generated

Suppose $G$ and $H$ are two free abelian groups of countable but not finite rank. Is a non-trivial homomorphism $f: G \rightarrow H$ necessarily injective?

Intuition says yes, but I am just worried about what happens when the rank is no longer finite. I am not comfortable with groups that are not finitely generated, as they rarely arise.

The obvious realisation of a countably generated free Abelian group is $\bigoplus_{i = 1}^\infty \mathbb{Z}$. That is, integer sequences $(x_i)_{i=1}^\infty$ with finite support (finitely many terms non-zero).
We can make a homomorphism from this set to itself, $(x_i)_{i = 1}^\infty \mapsto (x_1, 0, 0, \dots)$, collapsing all except the first term to zero, and this is pretty far from injective.