A set $A\subseteq\omega$ is called a cohesive set if $C$ is finite for each recursively enumerable set $W_e$, either $A\cap W_e$ is finite or $A\cap(\omega\setminus W_e)$ is finite. And a set $A\subseteq\omega$ is called simple if it is recursively enumerable and its complement has no infinite recursively enumerable subsets.

Let $C$ be a cohesive set. Then my question is, does there always exist a simple set $X$ such that $A\cap X$ is finite, i.e. all but finitely many elements of $A$ belong to $\omega\setminus X$?

A set $A\subseteq\omega$ is called a cohesive set if $C$ is finite for each recursively enumerable set $W_e$, either $A\cap W_e$ is finite or $A\cap(\omega\setminus W_e)$ is finite. And a set $A\subseteq\omega$ is called simple if it is recursively enumerable and its complement has no infinite recursively enumerable subsets.

Let $C$ be a cohesive set. Then my question is, does there always exist a simple set $X$ such that $C\cap X$ is finite, i.e. all but finitely many elements of $C$ belong to $\omega\setminus X$?