Infinite computably enumerable sets have computable injective enumerations

By [L1], choose a total computable enumerator $e:\mathbb{N}\to X$ with range exactly $A$, and let $E$ be a machine as in [L2] that computes it. On input $n$, simulate the computations of $E$ on $0,1,2,\dots$ one after another, keep the list of distinct output codewords already seen, and halt when the $(n+1)$st new codeword appears, outputting that codeword.

The resulting function $g:\mathbb{N}\to X$ is injective by construction, because each output is the next codeword not seen before. Every value of $g$ lies in $A$ because every output of $E$ lies in the range of $e$. Conversely, if $x\in A$, then some stage of the sequential scan in step 1.1 encounters the codeword $e_X(x)$ among the outputs of $E$, so that codeword is eventually appended to the distinct-output list. Hence the range of $g$ is exactly $A$.

Because $A$ is infinite, for each $n$ there are at least $n+1$ distinct elements of $A$. Step 2.1 shows that each of their codewords appears somewhere in the scan from step 1.1, so the $(n+1)$st new codeword is found after finitely many outputs of $E$. Therefore the machine of step 1.1 halts on every input $n$, and it computes a total injective function $g:\mathbb{N}\to X$ with range $A$.

Thus every infinite c.e. set has a computable injective enumeration.