Regular sequence

Let R be a regular local ring of dimension d and m = (x₁, x₂, ..., x_d) its maximal ideal. Then for each i, R / (x₁, x₂, ..., x_i) is a regular local ring of dimension d - i.

pf> Let S = R / (x₁, x₂, ..., x_d) and k = dim S. Clearly S is local. Since m is generated by d - i elements in S, d - i ≥ k. On the other hand, if m is minimal over (y₁, y₂, ..., y_l) in S, then m is minimal over (x₁, x₂, ..., x_i, y₁, y₂, ..., x_l) in R, so i + l ≥ d, i.e., l ≥ d - i. Since k is the smallest such number l, k ≥ d - i. Hence k = d - i. QED