Projective Noether normalization?

In commutative algebra the classic Noether normalization lemma says that every ring finitely generated over a field is a finitely generated module over a polynomial ring with coefficients in this field. The geometric interpretation of this statement is that if $X$ is an affine variety of dimension $n$ then there is a surjective finite map from $X$ to the affine $n$-space $\mathbb{A}^n$.

What about projective varieties? Does an analogous statement hold? That is, if $X$ is a closed subset of $\mathbb{P}^n$ of dimension $m$, is there necessarily a finite surjection $X \to \mathbb{P}^m$?

(Making my comment into an answer...)

Yes! Geometrically, take a general linear subspace $L$ in $\mathbb P^n$ of codimension $m+1$ disjoint from $X$, and $\Lambda \cong \mathbb P^m$ disjoint from $L$. Projection away from $L$ onto $\Lambda$ gives the finite map you want.

As Georges explained in the comments, a proper morphism with finite fibres is finite, as proved in Shafarevich, Basic Algebraic Geometry Vol. 1. This projection is proper because $X$ is projective, and has finite fibres because $X$ intersects any linear subspace of codimension $m$ containing $L$ in finitely many points. So this morphism is indeed finite.