I am trying to solve the exercise in Atiyah, that $\dim(A[X]) = \dim (A) + 1$ for $A$ noetherian.

The very beginning poses a problem, he states in the hint that:

for a prime of height $m$ we can choose $m$ elements in that prime such that the prime is a minimal prime over the ideal generated by those $m$ elements.

How might one prove that first statement? And is there an alternative approach to proving this?