Krull's height theorem

Statement
Let $$R$$ be a Noetherian commutative unital ring and $$x_1, x_2, \ldots, x_c$$ be elements in $$R$$. Let $$P$$ be minimal among primes containing all the $$x_i$$s. Then, the codimension of $$P$$ is at most $$c$$.

There is also a converse of this statement viz converse of Krull's height theorem.

Starting assumptions
Replacing $$R$$ by $$R_P$$ if necessary, we may assume that $$R$$ is a local ring with unique maximal ideal $$P$$.

In particular, we see that the ring $$R/(x_1,x_2,\ldots,x_c)$$ is a local Artinian ring with unique maximal ideal $$P/(x_1,x_2,\ldots,x_c)$$, hence $$P$$ is nilpotent modulo $$(x_1,x_2,\ldots,x_c)$$.

Main proof
Supose $$P_1$$ is a prime contained in $$P$$, with no primes between. Then, it suffices to show, inductively, that $$P_1$$ is minimal over an ideal generated by $$c-1$$ elements.