Exercise 17, § 7

Let $A$ be a local noetherian ring possessing a completely secant sequence of length $d = \dim(A)$, with the ideal generated by such a sequence and $f \in A$ an element integer over $a$ but not belonging to $a$ (for example $A = k[[X_1, ..., X_d]]$, where $k$ is a field, $a = (X_1^k, ..., X_d^k)$ and $f = X_1^{k-1}X_2$). Let $a' = a + fA$. Now prove that one has the inequality $e_{a'}(A) > \mathrm{long}(A/a')$.