This is not a great answer, but it was getting a bit long to be a comment, so I just made it community wiki instead. If anyone (with rep >= 100) wants to elaborate, feel free to do so in the answer itself rather than in the comments.
There is a technique for showing a closed subset $Z$ of an (irreducible) variety $X$ is all of $X$ that does not seem to have any analogue without using schemes. Let's suppose that $Z \subset X$ has a natural structure as a closed subscheme that comes from its definition (the induced reduced structure won't work here, as will become obvious). To show $Z = X$, it suffices to show that $Z$ contains a nonempty open set. If $z \in Z$, then to show $Z$ contains a neighborhood of $z$, it suffices to show that $Z$ contains every subscheme of $X$ supported on $z$--i.e., in a sense, $Z$ contains every infinitesimal neighborhood of $z$. A version of this is used in Mumford's book Abelian varieties to prove the Theorem of the Cube (II) in chapter III.