UTM Ideals Varieties and Algorithm - Chapter 1 Section 2 Exercise 6

Problem:

Solution:

Any single point is an affine variety $V(x_1 - a_1, x_2 - a_2, \cdots, x_n - a_n)$

Any finite subset of $k^n$ is a finite union of single points, which by applying lemma 2 repeatedly, is an affine variety.