Let $(A,\mathfrak{m})$ be a Noetherian local ring, $M\neq0$ a finite $A$-module. Suppose $d=\min\{{\operatorname{depth}A,\operatorname{depth}M\}}\geq1.$ Then does there always exists $a_1,\ldots,a_d\in\mathfrak{m}$ which is both an $A$-sequence and an $M$-sequence?
I came up with a very simple proof, but I'm a little surprised because I don't remember it being mentioned in any book. Can anyone confirm if this is true?