Solution
= Solution
The product $\prod_jG_j$ of the <finite groups> with their <discrete topology> is a <topological group> under coordinatewise multiplication and inversion. The compatibility equations defining $G=\varprojlim_jG_j$ are preserved by both operations, so $G$ is a subgroup. Their restrictions to the <subspace topology> on $G$ are continuous. Hence $G$ with its standard <inverse-limit topology> is a topological group.