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][1]. Also interesting: Nöbeling showed that the abelian group of bounded sequences in $\mathbb Z$ is free as an abelian group.


  [1]: http://mathoverflow.net/questions/37223/ultraproducts-of-finite-cyclic-groups/37249#37249