[Wikipedia](http://en.wikipedia.org/wiki/Slender_group) states that every free abelian group is slender. Where can I find a proof?


If this is not trivial, then I will also need a reference to use in my paper.



Every free abelian group is slender, why?
This follows fairly straightforwardly from the fact that every map from $A=\mathbb{Z}^\mathbb{N}$ to $\mathbb{Z}$ factors through a finite subproduct (let me call this "Specker's theorem"). Suppose $F$ is a free abelian group and $f:A\to F$ is a homomorphism. Since a subgroup of a free abelian group is free, we may assume $f$ is surjective. If $F$ is finitely generated, the conclusion follows from Specker's theorem, so we may assume $F$ is infinitely generated. Since $F$ is free, the map $f$ splits and so $F$ is a direct summand of $A$. But there are uncountably many homomorphisms $F\to\mathbb{Z}$, and hence uncountably many homomorphisms $A\to \mathbb{Z}$. This contradicts Specker's theorem.


