Let $R$ be a ring with unity and $M$ any unitary right $R$-module. We say that $M$ is finitely cogenerated if whenever $\lbrace A_\lambda \rbrace_{\lambda\in \Lambda}$ is a collection of submodules of $M$ with $\bigcap_{\lambda\in \Lambda} A_\lambda=0$ there exists a finite subset $\Lambda'\subset \Lambda$ such that $\bigcap_{\lambda\in \Lambda'} A_\lambda=0$. It's proven that a module is finitely cogenerated iff the intersection of any chain of nonzero submodules is nonzero.
For an essential submodule $B\subseteq M$, we write $B \subseteq^{\mathrm{ess}}M$.
Assume that whenever $K \subseteq^{\mathrm{ess}} N \subseteq M$ then $N/K$ is finitely cogenerated. How to prove that this implies that $M/A$ is finitely cogenerated for any submodule $A$ with $\mathrm{soc}(M) \subseteq A \subseteq M$, where $\mathrm{soc}(M)$ is the socle of $M$ which is the sum of all simple submodules (or, the intersection of all essential submodules).
I need any help. Thanks in advance.