My question is about the Ex. 4.9 page 31 in the book Algebraic Geometry by Robin Hartshone.
Let $X$ be a projective variety of dimension $r$ in $\mathbb{P}^n$, with $n\geq r+2$. Show that for suitable choice of $P\notin X$, and a linear $\mathbb{P}^{n-1} \subseteq \mathbb{P}^n$, the projection from $P$ to $\mathbb{P}^{n-1}$ (Ex. 3.14) induces a birational morphism of $X$ onto its image $X' \subseteq \mathbb{P}^{n-1}$.
An idea to prove this is to take a set of generators $\{x_1, \dots, x_n\}$ of $K(X)$ the the function field of $X$ and a subset $\{x_1, \dots, x_r\}$ that form a separating transcendence base (see Def. A on page 27). Then $K(X)$ is generated by a $k$- combination of $x_{r+1}, \dots, x_n$ and as $r\leq n-2$ there exist another combination of these elements which is not proportional to the first one. Now we choose a point $P \notin X$ and not in this plane.
But now, we have to prove that the projection from this $P$ is the one which induces the birrational morphism.
