Varieties strict inclusion infinitely long

Question

Let $F$ be a field, and consider the affine space $F^n$. Could there be affine varieties $V_1,V_2,\ldots\subseteq F^n$ such that $V_1\supset V_2\supset\cdots$, and no two are equal?

Answer 1

There is no infinite strictly decreasing sequence of closed subsets of $F^n$ because there is no strictly increasing sequence of ideals in the polynomial ring $F[T_1,...,T_n]$.
Indeed, that ring is noetherian by Hilbert's celebrated basis theorem.

Answer 2

You can create arbitrarily long chains: in $\mathbb{A}^1_k$ with coordinate ring $k[x]$, where $k$ is a field of characteristic $0$ (in particular infinite), $D(f)$ is an open affine subvariety for any $f \in k[x]$, and there is a chain of strict inclusions $D(x) \supset D(x(x-1)) \supset D(x(x-1)(x-2)) \supset \ldots$