Let $(R,\mathfrak{m})$ be a Cohen-Macaulay (CM) local ring of dimension $d>0$. Suppose that $M$ is a finitely generated CM $R$-module of $\dim_R M=i<d$.
Prove that $M$ is the homomorphic image of a CM $R$-module of dimension $i+1$.
My idea is to use induction on $d'\geq 0$. The case $i=0$ is pretty easy but I am stuck with the inductive step (I tried to find $U$ in the form $U'\oplus M$ but it seems to me that this does not work as $\mathrm{depth}(U'\oplus M)\leq \mathrm{depth} M$). I do not know if there are any ways to overcome this .