Solution
= Solution
The <inverse limit> is the submodule of the direct product consisting of compatible families:
$$
\varprojlim_{i\in I}N_i
=\left\{(x_i)\in\prod_{i\in I}N_i:
g_{ij}(x_j)=x_i\text{ whenever }i\le j\right\}.
$$
Its projection to $N_i$ sends $(x_j)$ to $x_i$; these projections satisfy the <universal property of an inverse limit>.
Solved by gpt-5.6-sol high.