Add induction argument

Thanks to Keenan Kidwell
http://stacks.math.columbia.edu/tag/07DV#comment-787

algebra.tex

@@ -14936,13 +14936,20 @@ \section{Regular sequences and depth}
 \begin{lemma}
 \label{lemma-regular-sequence-powers}
 Let $R$ be a ring. Let $M$ be an $R$-module.
-Let $f_1, \ldots, f_r \in R$ be $M$-regula.
-Then for $e_1, \ldots, e_r > 0$ the sequence $f_1^{e_1}, \ldots, f_r^{e_r}$
-is$M$-regular too.
+Let $f_1, \ldots, f_r \in R$ be $M$-regular.
+Then for $e_1, \ldots, e_r > 0$ the sequence
+$f_1^{e_1}, \ldots, f_r^{e_r}$
+is $M$-regular too.
 \end{lemma}
 
 \begin{proof}
-We will show that $f_1^e, f_2, \ldots, f_r$ is an $M$-regular sequence
+We will prove this by induction on $r$.
+If $r = 1$ this follows from the fact that a power of an $M$-regular
+element is an $M$-regular element. If $r > 1$, then by induction
+applied to $M/f_1M$ we have that $f_1, f_2^{e_2}, \ldots, f_r^{e_r}$
+is an $M$-regular sequence. Thus it suffices to show that
+$f_1^e, f_2, \ldots, f_r$ is an $M$-regular sequence if $f_1, \ldots, f_r$
+is an $M$-regular sequence. We will prove this
 by induction on $e$. The case $e = 1$ is trivial. Since $f_1$ is a
 nonzerodivisor we have a short exact sequence
 $$