A proof using Gröbner bases is in Using algebraic geometry by David A. Cox, John B. Little, Donal O'Shea, Theorem 2.1.
However, I was always sure that there should be (at least in the graded case) an inductive proof along the lines of Atiyah-Macdonald's proof of Hilbert--Serre theorem, namely by induction considering the 4-term exact sequence
$$0to K_ito M_ito M_{i+1}to L_ito0$$
where $K_n$ and $L_n$ are the kernel and the cokernel for the operator of multiplication by $x_n$ (these are modules over the polynomial ring in $n-1$ variables), but something escapes me at the moment, so I just leave it here as a wish....
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