 Coding of real sequences by ternary digits

Map each real into $(0,1)$, choose its canonical binary expansion, and flatten the resulting countable array of bits using a bijection $j:{\mathbb{N}}^2\to \mathbb{N}$. Interpret these bits as ternary digits zero and one. At a first differing position, the leading difference exceeds the sum of all possible tail differences, so this is an injection from real sequences to a bounded real interval. The bijection $j(n,k)=(n+k-2)(n+k-1)/2+n$ lists successive finite diagonals.