Alternative proofs of Schwartz–Zippel lemma
As a followup to Per Vognsen's answer, Dana Moshkovitz's proof already suggests a really easy proof for only a slightly weaker version of the Schwartz-Zippel Lemma that, I think, suffices for most applications. 

Let $f : \mathbb{F}^n \to \mathbb{F}$ be a nonzero polynomial of degree $d$, where $\mathbb{F}$ is a finite field of order $q$, and let $x \in \mathbb{F}^n$ be a point such that $f(x) \neq 0$.  There are $(q^n-1)/(q-1)$ many distinct lines passing through $x$ such that they partition $\mathbb{F}^n-\{x\}$.  The restriction of $f$ to each of these lines is a degree $d$ univariate polynomial, which is nonzero, because it is nonzero at $x$, and so, has at most $d$ zeros.  Thus, the total number of zeros of $f$ is at most $d(q^n-1)/(q-1)$.  Schwartz-Zippel, for comparison, gives the stronger upper bound of $d q^{n-1}$.

Given this proof's easiness, I'm sure it's folklore; if not, it should be :)  I'd appreciate if someone could provide a reference.