Let's say I have two functions $f,g: \mathbb{Z} \mapsto \{0,..,9\}$
and let $ c_0(x) = f(x) + g(x)$ and $ c_{i+1}(x) = ( c_i(x) \mod 10) + \lfloor \frac{c_i(x - 1)}{10} \rfloor$.
Does $(c_{i})_{i\geqslant 1}$ eventually converge to a function $c_\infty$?
Notice $$c_1(x) \leq 9 \quad \forall x\in \mathbb{Z}\implies c_1(x)=c_2(x)=c_3(x)= \cdots=c_{\infty}.$$
Am I missing something ?