algebraic geometry exercise: infinite subset is dense

Question

A hypersurface $C \subset \mathbb{A}^2$ we call a plane curve. Show that any infinite subset of an irreducible plane curve $C \subset \mathbb{A}^2$ is dense in $C.$

Note. We call hypersurface the zero set $Z(f)$ of a non-constant polynomial $f \in k[x_1,\ldots,x_n],$ and we don't know the dimension of a hypersurface, nor that $\dim \mathbb{A}^n=n.$

Edit: $k$ algebraically closed.

Answer 1

Hint: Every plane curve is homeomorphic (though not isomorphic!!) to $\mathbb{A}^1$ (why?). So it suffices to show that every infinite subset of $\mathbb{A}^1$ is dense in $\mathbb{A}^1$. This comes down (why?) to the fact that the only polynomial in $k[x]$ having infinitely many zeros is the zero polynomial.

Edit: I should add that I have $k$ algebraically closed in mind.