a) Prove that there is a curve $\alpha$ and sequence $\{x_n\}_{n \geq 0}$ in $\Bbb R^2$ such that $wind_{xn}(\alpha)=n$ for all $n \geq 0$. ($wind_{x_n}(\alpha)$ refers to the winding number of $\alpha$ around $x_n$ )
b) Also prove that $\{x_n\}_{n \geq 0}$ is necessarily bounded and that all of its limit points lie in $\alpha$.
I think I can somewhat manage part b) (my attempt is below), but how do you prove the existence of $\alpha$ and $\{x_n\}_{n \geq 0}$?
b) I'd say $\{x_n\}_{n \geq 0}$ must be bounded so that $\alpha$ can be closed, and $wind_{xn}(\alpha)=n$ for all $n \geq 0$ means that $x_n \neq x_m$ for $n \neq m$ that there are indeed limit points of $\{x_n\}_{n \geq 0}$. If p is a limit point of $\{x_n\}_{n \geq 0}$ then it will also be a limit point of $\alpha$, meaning that $p \in \alpha$, is that correct?