Let $R$ be a Noetherian local ring with maximal ideal $\mathfrak{m}$. Is it true that for a prime ideal $\mathfrak{p}$ in $R$, and $R'=R/\mathfrak{p}$, we have an exact sequence
$$ 0\to \mathfrak{p}/\mathfrak{m}^n\to R/\mathfrak{m}^n \to R'/\mathfrak{m}'^n\to 0 $$
for any $n$? I want to prove $l(R/\mathfrak{m}^n)\geq l(R'/\mathfrak{m}'^n)$ where $l$ is the length of the module. Is this method okay?