On algorithmical sequences belonging to the initial class of Grzegorczyk hierarchy

It is proved that for any recursive sequence $f$ there exists a sequence $g$ of Grzegorczyk class $E^0$ such that $g(0),g(1),\dots$ is obtained from $f(0),f(1),\dots$ by replacing some members $f(i)$ by finite sequences $f(i),\dots,f(i)$.
This implies that every recursively convergent recursive sequence of rational numbers can be represented by a functions from $E^0$.