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Let $k$ be a field, $R = k[x_1,\dots,x_n]$ the polynomial ring, $\mathfrak m = (x_1,\dots,x_n)$ and $M$ a finitely generated graded $R$-module. How can we see that $\operatorname{reg}(\mathfrak mM) \le \operatorname{reg}(M) + 1$, where $\operatorname{reg}(\cdot)$ denotes Castelnuovo-Mumford regularity?

Remark: i can see that $\operatorname{reg}(\mathfrak m)=1$.

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Since $M/\mathfrak mM$ is an $R$-module of finite length, $\operatorname{reg}(M/\mathfrak mM)=\max\{i:(M/\mathfrak mM)_i\ne 0\}$; see here. Now, since $\operatorname{reg}(M)\ge\max\{j:\beta_{0j}(M)\ne 0\}$, it follows that $\operatorname{reg}(M)\ge \operatorname{reg}(M/\mathfrak mM)$.

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