= Solution
The quotient maps define a continuous homomorphism
$$
\Phi:G\longrightarrow\varprojlim_{U\in\mathcal U}G/U.
$$
Its kernel is $\bigcap_{U\in\mathcal U}U=\{1\}$, because $\mathcal U$ is a <neighborhood basis> and $G$ is <Hausdorff>. Thus $\Phi$ is injective. The standard compactness argument for <inverse limits> makes it surjective: a compatible family of cosets has the <finite intersection property>, and the corresponding closed cosets in compact $G$ have nonempty intersection. Finally, a continuous bijection from compact $G$ to the Hausdorff inverse limit is a <homeomorphism>. Hence
$$
G\cong\varprojlim_{U\in\mathcal U}G/U
$$
as <topological groups>.
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