Tell if the subset $E=\{(x,y) \in \mathbb R^2 | y^2-xy-x^3=0\}$ of $(\mathbb R^2,\epsilon_1)$, with $\epsilon_1 $the usual topology is compact.
The solution I was given:
For each $n \in \mathbb N$, the equation $-x^3-nx+n^2=0$ has at least one real root $x_n$. Then, the unbounded sequence $(x_n,n)$ is contained in E, from which E is not compact.
I really don't understand it. Can someone please explain
1) why the equation has at least one real root, is it Descartes' rule?
2) how do we know the sequence is unbounded, could they all be contain in a bounded region like inside a circle or something, how to tell?
If you fix $y$, you have a cubic in $x$. Then a real root exists, because the function $f(x)=-x^3-xy+y^2$ is continuous and has $\lim_{x\to-\infty}f(x)=+\infty$, while $\lim_{x\to\infty}f(x)=-\infty$. Or you can argue that complex roots appear in pairs when the coefficients are real.
In any case, now you can take $y=n$, and you get a root $x_n$. That is, $n^2-nx_n-x_n^3=0$. So you get points in your set of the form $(x_n,n)$. The distance to the origin is $\sqrt{x_n^2+n^2}\geq n$, so the sequence is unbounded.