This is exercise 2.21.
Let $k$ be field, $k[X_1, \ldots, X_n]$ be polynomial ring. $S \subseteq k$ have $dn+1$ elements. Let $d$ be the highest power of any variables appearing in $f$. Proceeding by induction find $a_1, \ldots, a_n \in S$ such that $f(a_1,\ldots, a_n) \not=0 $
Why do I only need $d+n$(in fact $d+1$ if we allow repeated) elements instead of $dn+1$ elements? Let me illustrate in the $n=2$ case.
$f(x_1,x_2) = \sum_{i=0}^d g_i x_1^i$ where $g_i \not= 0$ for some $i$. So $g_i \in k[x_2]$ and has degree at most $d$. Hence, exists $a_2 \in S$, such that $g_i(a_2) \not= 0$. Now consider $f(x_1, a_2)= \sum g_i(a_2)x_1^i$. $f(x_1, a_2)$ has degree $\le d$ and nonzero. From $n=1$ case, $S\setminus \{ a_2 \}$, exists $a_1$, such that $f(a_1, a_2) \not= 0$.
Something must have gone wrong in my proof.