Let us consider $A$ a regular local noetherian ring that contains an algebraically closed field $k$ and whose residue field is again $k$ (eg. the local ring of a nonsingular variety over $k$ at a closed point). We denote by $\mathfrak{m}$ the maximal ideal of $A$.
Let $\overline{t_1},\ldots,\overline{t_r}$ be a basis of the $k$-vector space $\mathfrak{m}/\mathfrak{m}^2$, where $r=dim(A)$. Let $t_1,\ldots,t_r \in \mathfrak{m}$ be representatives of $\overline{t_1},\ldots,\overline{t_r}$. By Nakayama's lemma, we know that the $t_i$'s form a minimal set of generator of $\mathfrak{m}$.
Does it follow that for each $i$, $t_{i+1}$ is not a unit and not zero divisor in $A/(t_1,\ldots,t_i)$ ? If so, could someone please provide me a proof of this statement or a reference for it ?
I thank you very much for your help.
NB: For context, this interrogation rises from the proof of Proposition III.10.4 in Hartshorne's book.