The inverse limit of finite discrete groups is a closed topological subgroup of the full product

For each $i\le j$, define ${\mathrm{\Psi }}_{ij}:{\prod }_k{G}_k\to G_i\times G_i,{\mathrm{\Psi }}_{ij}((g_k))=(g_i,{\phi }_{ij}(g_j))\mathrm{.}$ Because every factor is discrete, $G_i\times G_i$ is discrete, and the diagonal ${\mathrm{\Delta }}_i=\{(x,x):x\in G_i\}$ is closed. A tuple is compatible at the pair $(i,j)$ exactly when it lies in ${\mathrm{\Psi }}_{ij}^{-1}({\mathrm{\Delta }}_i)$. Therefore $L={\bigcap }_{i\le j}{\mathrm{\Psi }}_{ij}^{-1}({\mathrm{\Delta }}_i)$ is closed in the ambient product.

By [L1], $L$ is a subgroup of the product group. The product group operations are continuous coordinatewise, and restricting continuous maps to a subspace preserves continuity. Hence $L$ is a topological subgroup in the sense of [F1].

So $L$ is a closed topological subgroup of the product of the finite discrete groups.