= Solution
Suppose $A=\operatorname{ran}h$ for a <total computable function> $h$. Effectively enumerate the words as $w_0,w_1,\ldots$. On input $x$, compute
$$
h(w_0),h(w_1),\ldots
$$
in order and halt as soon as one equals $x$. Every individual computation terminates because $h$ is total. The search halts exactly for $x\in\operatorname{ran}h=A$, so $A$ is <computably enumerable set>[computably enumerable]. This proves (iv)$\Rightarrow$(i) and completes the equivalence recorded by the <domain and range characterizations of a nonempty computably enumerable set>.
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