I am studying the book "Local Fields" from Serre. At the beginning of chapter III the setting is follows: $A$ denotes a Dedekind domain, $K$ its field of fractions and $V$ is a finite-dimensional $K$-vector space. A lattice of $V$ is an $A$-submodule of $V$ that is finitely generated, and that spans $V$.
Then he continues: if $X_1,X_2$ are two lattices of $V$; if $X_2\subseteq X_1$, then $X_1/X_2$ is a module of finite length. What is an easy argument to see this?
Thanks in advance!
Each $x\in X_1$ can be written as a $K$-linear combination of elements of $X_2$. If we multiply away denominator, we see that $cx\in X_2$ for suitable $0\ne c\in A$. If $x_1,\ldots,x_n$ generate $X_1$ and lead to corresponding factors $c_1,\ldots, c_n$ with $c_ix_i\in X_2$, then let $c=c_1\cdots c_n$ and find that $cX_1\subseteq X_2$. Then $X_1/X_2$ is a submodule of $(A/(c))^n$. Can you take it from here?