# Is it possible to refine any ascending/descending chain to a composition series for a module?

As the title asks: Is it possible to refine any ascending/descending chain to a composition series for a module? I am trying to prove that any module has a finite composition series iff it is both artinian and noetherian. (Please do not provide answer). And I had a thought I was unsure about. For any arbitrary chain

$$A_1 \subset A_2 \subset \dots \subset A_k \subset A_{k+1} \subset ...$$

can this be refined to a composition series for the Module containing them?