Bézout's theorem says that given $f, g$ homogeneous polynomials over $\mathbb{CP}^2$ with degrees $n, m$ respectively, the number of intersections of their zero sets is exactly $nm$, counted with multiplicities.
I know of two proofs that can be found in modern references. One uses the more classical machinery of resultants (in which intersection multiplicity is defined as the exponent of a particular term in a particular resultant), and the other uses the more modern machinery of local rings (in which intersection multiplicity is defined as the dimension of a particular vector space).
Wikipedia alludes to a third, less rigorous proof that goes somewhat along these lines: The intersection multiplicity of $f$ and $g$ at $p$ can be computed by applying small perturbations to the coefficients of $f$ and $g$, then counting the maximum possible number of intersections in a small neighborhood of $p$. In doing so, we reduce the problem to counting intersections of generic polynomials, which is easy (but I don't see why it's easy).
Apparently, this is Bézout's original proof, so I tried reading a translation of his original book General Theory of Algebraic Equations. But I have a difficult time parsing the older language. Is anyone able to help me write down the correct definitions and sketch the basic steps of this argument? Thanks!