= Solution
An <inverse system> of <topological spaces> over a <directed set> $I$ consists of spaces $X_i$ and continuous maps $f_{ij}:X_j\to X_i$ for $i\leq j$, with $f_{ii}=1$ and $f_{ik}=f_{ij}f_{jk}$. Its <inverse limit> is a space $X$ with compatible continuous projections $\pi_i$ such that every compatible family of continuous maps $u_i:Y\to X_i$ factors uniquely through a continuous map $u:Y\to X$. To construct it, take
$$
X=\{(x_i)\in\prod_{i\in I}X_i:f_{ij}(x_j)=x_i\text{ whenever }i\leq j\}
$$
with the <subspace topology> inherited from the <product topology>. The map $u(y)=(u_i(y))$ has values in $X$ and is continuous by the definition of the <product topology>; its coordinates force uniqueness. If $X'$ is another <inverse limit>, the two universal properties give continuous maps $X\to X'$ and $X'\to X$ whose composites have all the same projections as the identity. Uniqueness makes these composites the identity. \b[The inverse limit exists and is unique up to a unique projection-preserving homeomorphism.] An arbitrary <inverse limit> may be empty; nonemptiness requires additional hypotheses.
Use the usual Hausdorff convention for a <topological group>. An <inverse limit> of finite discrete <groups> is a closed subgroup of their product: each compatibility condition is closed because the finite factors are <Hausdorff spaces>. The <Tychonoff theorem> gives <compactness>, and distinct tuples are separated by a clopen coordinate cylinder, so the limit is <totally disconnected>. Coordinatewise multiplication and inversion are continuous, giving a <profinite group>.
Conversely, let $G$ be a <compact Hausdorff space> and a <totally disconnected> <topological group>. We use the standard topological fact that a <compact Hausdorff space> that is <totally disconnected> has a basis of <clopen sets>. Given an identity neighbourhood, choose a clopen identity neighbourhood $U$ inside it. <Compactness> of $U$ and continuity of multiplication give a symmetric identity neighbourhood $V$ with $VU\subseteq U$: cover $U$ by finitely many neighbourhoods on which multiplication by a sufficiently small identity neighbourhood stays inside $U$, then intersect those identity neighbourhoods. Symmetry gives $vU=U$ for $v\in V$. Consequently $H=\{g:gU=U\}$ is a subgroup containing $V$, hence an <open subgroup>, and $H\subseteq U$ since $1\in U$. Its open <cosets> cover the compact group, so there are finitely many. The <normal core> $N=\bigcap_{g\in G}gHg^{-1}$ is a finite intersection of open subgroups, hence an <open normal subgroup> contained in $U$. Thus the open normal subgroups form an identity-neighbourhood basis and have trivial intersection.
The canonical <group homomorphism> $G\to\varprojlim_NG/N$ is continuous and injective. For a compatible tuple of quotient elements, its inverse-image cosets in $G$ are closed and have the <finite intersection property>: a finite family is refined by the quotient for the intersection of its normal subgroups. <Compactness> gives an element in all those cosets, proving surjectivity. A continuous bijection from a <compact space> to a <Hausdorff space> is a <homeomorphism>. \b[The compact totally disconnected and inverse-limit definitions of a profinite group agree.]
Index the finite nonempty sets $\mathcal P(N)$ by open normal subgroups, with the transition from $M$ to $N$ when $M\subseteq N$ given by image under $G/M\to G/N$. Images of <Sylow subgroups> under surjections of finite <groups> are <Sylow subgroups>. These transitions are surjective: if $S\leq G/N$ is a <Sylow subgroup>, choose a <Sylow subgroup> in its full inverse image in $G/M$. It maps onto $S$, and its order has the full $p$-part of $|G/M|$, so it is a <Sylow subgroup> of $G/M$ too. The transitions respect composition, giving an <inverse system> of finite sets.
Its <inverse limit> is nonempty. Indeed, finitely many compatibility conditions can be satisfied by choosing one <Sylow subgroup> in a quotient refining all the involved quotients and taking its images; <compactness> of the product of the finite sets then gives a compatible family $(P_N)_N$. Define
$$
P=\bigcap_N\pi_N^{-1}(P_N).
$$
This is a closed subgroup of $G$. Its image in $G/N$ is exactly $P_N$: for a prescribed $s\in P_N$, each finite collection of conditions defining $P$ together with $\pi_N(g)=s$ is satisfiable in a common refining quotient, because the transitions on the chosen <Sylow subgroups> are surjective. The <finite intersection property> again supplies $g\in P$. Hence $P\cong\varprojlim_NP_N$ is a <pro-p group> and $PN/N=P_N$ for every $N$.
Finally, if a <pro-p subgroup> $Q$ contains $P$, its image in every finite quotient is a <p-group> containing the <Sylow subgroup> $P_N$, so that image equals $P_N$. Thus every $q\in Q$ belongs to every $\pi_N^{-1}(P_N)$, whence $Q\subseteq P$. \b[The constructed $P$ is a maximal pro-p subgroup.]
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