The profinite completion is the inverse limit of the finite quotients G over N

Let $G$ be an abstract group. Its profinite completion is the inverse limit

where the index set is the directed family of finite-index normal subgroups, ordered by reverse inclusion, and the transition maps are the natural quotient homomorphisms $G\mathrm{/}{N}^{\mathrm{\prime }}\to G\mathrm{/}N$ whenever $N^{\mathrm{\prime }}\subseteq N$ (A directed set and an inverse system of groups indexed by it, The profinite topology on a group uses finite-index normal subgroups as an identity-neighbourhood basis, The quotient group $G\mathrm{/}N$ and coset product $(gN)(hN)=ghN$).

The indexing family is nonempty because it contains $G$. It is directed because $N_1\cap N_2$ is normal and has finite index: the diagonal map embeds $G\mathrm{/}(N_1\cap N_2)$ into the finite group $G\mathrm{/}{N}_1\times G\mathrm{/}{N}_2$, and in the reverse-inclusion order $N_1\cap N_2$ is a common upper bound of $N_1$ and $N_2$.

Give each quotient the discrete topology and $\widehat{G}$ the inverse-limit topology of The inverse limit of finite groups carries the subspace topology from the product of discrete factors. With this topology it is a topological group by The inverse limit of finite discrete groups is a closed topological subgroup of the full product. Since it is itself an inverse limit of finite discrete groups, it is profinite by A profinite group is a topological group isomorphic to an inverse limit of finite discrete groups.