A Gröbner-based treatment of elimination theory for affine varieties

We present alternative proofs for the well-known Extension and Closure Theorems of elimination theory for affine varieties. Assuming knowledge of Gröbner basics, the proofs are quite short and elementary. They also hardly resort to any machinery from more classical algebraic geometry. Instead, they are designed to be easily presentable along with the (Gröbner basis) techniques likely to be used in applications of the results.