I've been stuck on the following question from dimension theory in commutative algebra.
Let $(A,m)$ be a local ring and $M$ a finitely generated $A$-module.
Given $x_1,...,x_r \in \mathfrak{m}$, prove that $\dim(\frac{M}{(x_1,...,x_r)M}) \geq \dim (M) - r$, with equality holding if and only if {$x_1,...,x_r$} is a part of a system of parameters for $M$.
Now I can show that if $A$ is a $\mathbf{regular}$ local ring, then $\frac{A}{(x_1,...,x_r)A}$ is a regular local ring with dimension $\dim (A) - r$ if and only if {$x_1,...,x_r$} is a part of a system of parameters for $A$. But I don't know how to show this for the given case. I also cannot show the inequality. I haven't been able to find a proof for this, so I'd appreciate any help with this!