For the purposes of teaching my elementary course in algebraic geometry I am looking for a reference (or notes) that contains a complete proof of a higher-dimensional weak Bezout theorem. I only want to learn about a proof that is based on the approach when dimensions and degrees of projective varieties are introduced via Hilbert polynomials.
Weak Bezout. Let $F_1,...,F_n$ be homogeneous polynomials of degrees $d_1,...,d_n$ such that hypersurfaces $F_1=0$,..., $F_n=0$ have only finite number of intersections in $\mathbb P^n_K$ (with $K$ algebraically closed). Then the number of intersections is at most $d_1\cdot...\cdot d_n$.
Justification of the question. Of course there is a large amount of proofs of this statement in many books on algebraic geometry (Shafarevich, Harris, Hasset, ect.) but these proofs usually come after 200 pages of text, while I want an honest proof that is contained in a complete set of notes of 40 pages (this will be the maximal length of my notes and I don't want to spend more than 10 pages on higher dimensional Bezout). Also these proofs often develop dimension theory basing on transcendence degree of the field of rational functions and I don't want to use this approach, (since it will require 2-3 additional lectures which I don't have time to give). I know one place where the approach that I want to use (namely everything is based on Hilbert polynomials) is taken, these are the notes of Manin (end of 60ties). Unfortunately it seems to me that there is a problem with the proof he proposes. Basically everything works if $F_1,...,F_n$ form a regular sequence, but to understand why in the condition of the theorem $F_1,...,F_n$ do form a regular sequence is left in the notes as something "not hard to do"... In the book of Hasset, it seems to me there is a similar problem (i.e. it is not explained why $F_1,...,F_n$ form a regular sequence if the number of intersections of hypersurfaces is finite). I hope there is a nice complete proof somewhere...
In other words, what will completely satisfy me is a short proof of the following statement: If the number of intersections of $F_i=0$ is finite, then $F_i$ form a regular sequence. Is there a short proof of this statement?
PS. I think that the answer of Sandor shows that probably there is no "magic easy" solution to the question (if one sticks to Hilbert polynomial approach instead of using higher-dimensional resultants), so I decided to accept it.