The Gijswijt’s sequence throughout the integers

We define the Gijswijt’s sequence as follows : $\ \left\{ \begin{array}{ll} x_0 = 1,\\ x_{n+1}= 10*x_{n} + max_{(i,j )\in \mathbb{N}^{2}, \: i+j \leq n}(Card\left\{\lfloor \frac{x_{i}}{10^{i}} \rfloor = \lfloor \frac{x_{i+j}}{10^{i+j}} \rfloor \right\})\end{array} \right.$
In other words, for a term $x_n$ of the sequence, the next term appends to its right the maximum length of consecutive blocks of identical numbers.

Show that every positive integer can be found in this sequence, i.e. that $\forall k \in \mathbb{N}, \exists n \in \mathbb{N} \text{ such that } max_{(i,j )\in \mathbb{N}^{2}, \: i+j \leq n}(Card\left\{\lfloor \frac{x_{i}}{10^{i}} \rfloor = \lfloor \frac{x_{i+j}}{10^{i+j}} \rfloor \right\}) =k$.