A binary language is recognizable if and only if it is computably enumerable

Assume first that $L$ is c.e. If $L=\mathrm{\varnothing }$, then the machine that immediately rejects every input recognizes $L$. Otherwise choose a total computable enumerator $e:\mathbb{N}\to \{0,1{\}}^{\ast }$ with range $L$ as in [L2]. On input $x$, search through $n=0,1,2,\dots$ and compute $e(n)$ until $e(n)=x$; if this occurs, accept. For $x\in L$ the search eventually finds a witness $n$, and for $x\mathrm{\notin }L$ it never accepts. Thus $L$ is recognizable by [L1].

Assume conversely that $L$ is recognizable. If $L=\mathrm{\varnothing }$, then [L2] already says that $L$ is c.e. So assume $L\mathrm{\ne }\mathrm{\varnothing }$, and let $M$ be a recognizer for $L$. By dovetailing the simulations of $M$ over binary words in shortlex order, fix one accepted word $a_0\in L$.

Fix a standard total computable bijection $n\mapsto (i,t)$ from $\mathbb{N}$ onto $\mathbb{N}\times \mathbb{N}$, and let $w_i$ be the $i$th binary word in shortlex order. Define $e(n):=\{\begin{array}{ll}{w}_i,& \text{if }M\text{ accepts }{w}_i\text{ within }t\text{ steps},\\ a_0,& \text{otherwise.}\end{array}$ This is a total computable function $\mathbb{N}\to \{0,1{\}}^{\ast }$. Every output of $e$ lies in $L$, and if $w\in L$ then $w=w_i$ for some $i$ and $M$ accepts $w_i$ in some finite number of steps $t$, so $e(n)=w$ for the paired input with coordinates $(i,t)$. Hence the range of $e$ is exactly $L$, so [L2] shows that $L$ is c.e.

Steps 1.1 and 2.1 prove the equivalence.