Let $f_1,f_2,\cdots$ be an infinite sequence of polynomials in $k[x_1,\cdots,x_n]$ and let $V(f_1,f_2,\cdots)=\{(a_1,\cdots,a_n)\in k^n:f_i(a_1,\cdots,a_n)=0$ for $i=0,1,\cdots\}$. Show that there is some finite $m$ such that $V(f_1,\cdots,f_m)=V(f_1,f_2,\cdots)$.
I think Hilbert basis theorem may help.
Yes, the vanishing set $V(T)$, where $T$ is a subset of $k[x_1,\ldots, x_n]$ equals $V(J)$, where $J$ is the ideal generated by $T$. But $k[x_1,\ldots,x_n]$ is noetherian, so that $J$ can be generated by finitely many polynomials