= Solution
Choose a countable generating family $(n_k)$ for $N$. For each $k$, the expression of $n_k$ in the <basis of a module> $E$ has a finite set $H_k$ of nonzero coefficients, its <coordinate support in a free module>. Let
$$
E'=\bigcup_kH_k.
$$
By the <countable union of countable sets> property, $E'$ is a <countable set>. Every $n_k$ belongs to $\langle E'\rangle$, so all finite $R$-linear combinations of the $n_k$ do too. Therefore
$$
\boxed{N\subseteq\langle E'\rangle\quad\text{for a countable }E'\subseteq E}.
$$
Here and below, <countable> includes finite. The same construction applies to a finite or empty generating family. This is the <countable basis support lemma>.
Back to article page
