= Solution
The <Weak Mordell-Weil theorem> asserts that $E(K)/mE(K)$ is finite for an <elliptic curve> over a <number field> $K$ and an <integer> $m\ge2$. The role of (i) is to make division by $m$ possible in the local <formal group of an elliptic curve> whenever the <residue characteristic> does not divide $m$.
Let $v$ be a finite place with <good reduction of an elliptic curve> and $v\nmid m$. The <kernel of reduction of an elliptic curve> is parametrized by $t=-x/y$ in the <maximal ideal> of the local <valuation ring>. Multiplication by $m$ is represented by an integral <formal power series>
$$
[m](T)=mT+\text{terms of degree at least }2.
$$
Its linear coefficient is a local <unit>. Part (i) supplies an integral compositional inverse. Both series converge on the <maximal ideal>, so multiplication by $m$ is a <bijection> on the reduction kernel, over the <local field> and over every finite <unramified extension>.
This implies <unramified division torsors at good primes>. For $P\in E(K)$, first divide its reduction by $m$ over the algebraic closure of the <residue field>. Multiplication by $m$ on the reduced curve is <surjective> and separable, so the chosen division point is defined over some finite residue extension. Smooth lifting gives a point $Q_0$ over the corresponding unramified local extension with the required reduction. The difference $P-mQ_0$ lies in the reduction kernel; the inverse series supplies $Q_1$ with $mQ_1=P-mQ_0$. Thus $Q=Q_0+Q_1$ divides $P$ over an <unramified extension>. Applying the same reasoning to all reduced $m$-torsion points lifts all of $E[m]$, so every point of $[m]^{-1}P$ lies in the maximal <unramified extension>. This excludes ramification outside a fixed <finite set> $S$ consisting of <primes> of bad reduction and <primes> over $m$.
For completeness, the remaining global finiteness step in the <Kummer-theoretic proof of the weak Mordell-Weil theorem> is as follows. Choose $Q\in E(\overline K)$ with $mQ=P$. The <cocycle> $\sigma\mapsto\sigma Q-Q$ defines the <Kummer map of an elliptic curve>
$$
E(K)/mE(K)\hookrightarrow H^1(K,E[m]).
$$
Changing $Q$ by an $m$-torsion point changes the <cocycle> by a <coboundary>. A zero class makes $Q$ rational after that change, so the map is injective.
Let $L=K(E[m])$, fixed independently of $P$. Since the torsion is rational over $L$, the <cocycle> restricted to its <Galois group> is a homomorphism into $E[m]$. Consequently $L(Q)/L$ is Galois of degree at most $m^2$, contains the entire division fibre, and is unramified outside $S$. Hence $[L(Q):K]\le m^2[L:K]$. There are only finitely many extensions of bounded degree unramified outside a fixed <finite set>: local extensions of bounded degree have bounded <field discriminant> exponents, and the <Hermite–Minkowski theorem> then applies to the bounded global <field discriminants>. Their finite Galois compositum $M$ contains every such division fibre; each $L(Q)$ is stable over $K$ because it contains $E[m]$ and the whole fibre above the rational point $P$. All the Kummer classes consequently factor through the fixed <finite group> $\operatorname{Gal}(M/K)$ with values in the fixed <finite group> $E[m]$. There are only finitely many such <cocycles>, proving
$$
\boxed{E(K)/mE(K)\text{ is finite}.}
$$
Thus (i) provides the local division step that bounds ramification; the global arithmetic finiteness completes the argument.
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