 Coherent coinfinite injections into omega

There are injections $e_{\alpha }:\alpha \to \omega$ for $\alpha <{\omega }_1$, each with coinfinite range, with $e_{\alpha }\upharpoonright \beta$ equal to $e_{\beta }$ outside a finite set. At countable limits, extend partial functions along a cofinal sequence, repair finitely many collisions, and reserve a fresh omitted natural number at each step. The reserved sequence makes the union coinfinite and permits all later successor steps.