I have been reading the book an Invitation to Algebraic Geometry. And I meet a problem in section 3.2.
Exercise 3.2.1 Show that every projective variety in $\mathbb{P}^n$ is compact in the induced Euclidean topology. Show that projective varieties are compactifications of affine varieties, in both the Zariski topology and, more significantly, in the Euclidean topology.
I am confused about that projective varieties are compact.
For example, $\mathbb{V}(x^2+y^2-z^2)$ in $\mathbb{P^n}$. Consider the points like $(1, \sqrt{t^2-1}, t)$. It's a curve in $\mathbb{A}^2$. How could it be compact in the induced Euclidean topology?