Past exam of the mathematics course of the University of Cambridge / 2023 / ii / Paper 2 / 4I / b / iv

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