Descending sequence for module with finite length

Question

Let $(R,\mathfrak{m})$ be a Noetherian local ring and $M$ be a finite $R$-module with finite length. Then the descending sequence $$M\supseteq\mathfrak{m}M\supseteq\mathfrak{m}^2M\supseteq\cdots\supseteq\mathfrak{m}^nM\supseteq\cdots$$ must stable for $n$ big enough. My question is will it always descend to zero? Maybe this is very simple but I just can't see it. Hope someone would help me. Thanks in advance!