It is known (this is Specker's theorem) that the natural map $$\iota :\bigoplus_{n \in \mathbb N} \ \mathbb Z \to Hom_{\mathbb Z} \left( \prod_{n \in \mathbb N} \mathbb Z,\mathbb Z \right)$$ is an isomorphism of abelian groups.
In particular, $\prod_{n \in \mathbb N} \mathbb Z$ cannot be a free abelian group. The crucial part of the proof appeared here. Also interesting: Nöbeling showed that the abelian group of bounded sequences in $\mathbb Z$ is free as an abelian group.