Initial-form lower bound for ideal generators (${\dim }_{k}I/\mathfrak{m}I\text{ bounds the generator count}$)

Let $\mathfrak{m}$ be a maximal ideal of a ring $A$, $k=A/\mathfrak{m}$, and $I\subseteq {\mathfrak{m}}^d$. Since $\mathfrak{m}I\subseteq {\mathfrak{m}}^{d+1}$, elements of $I$ with linearly independent classes in ${\mathfrak{m}}^d/{\mathfrak{m}}^{d+1}$ have independent classes in $I/\mathfrak{m}I$. Any generating set must span this latter vector space. Thus $r$ independent degree-$d$ initial forms give a lower bound of $r$ generators, without assuming that height equals the generator count.