By diagonalization, it is possible to construct a real number $r \in [0,1]$ such that for every rational $q \in [0,1]$, there exists an index $i \in \mathbb{N}$ such that $r_i \neq q_i$ (where $x_i$ is the $i$'ith digit in $x$).  

Can we make a stronger claim, and construct a real number $r \in [0,1]$ such that for every rational $q \in [0,1]$ there exists an index $i \in \mathbb{N}$ such that for **every** $j \geq i$, $r_j \neq q_j$?