Prove that $K=\{(x,y,z)\in \Bbb{R}^3\ :\ x^2+yz=x+1\}$ is not compact

Question

Let $K=\{(x,y,z)\in \Bbb{R}^3\ :\ x^2+yz=x+1\}$

Show that $K$ is not compact

Answer 1

$K$ is "clearly" closed, as the solution set of a continuous function. So the only way it can fail to be compact is to be unbounded.

Now check that $(0, n , \frac{1}{n}) \in K$ for all $n$ and its norm is $>n$.