Injective function

Map each real into $(0,1)$ by a fixed bijection, choose its canonical binary expansion, and alternate the two bit sequences as ternary digits $0$ and $2$. If two such outputs first differ at position $k$, the leading difference has size $2\cdot 3^{-k}$ and the remaining tail has size at most $3^{-k}$, so they are unequal. This gives an explicit injection pairing two reals. Iteration encodes every fixed finite real tuple without invoking ambiguous decimal expansions.

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.