I am interested in seeing why the statement
(1) "If $Y$ is any variety and $Z$ a closed subset of $\mathbb{P}^n \times Y$, then the projection of $Z$ on $Y$ is closed."
implies the statement
(1) "A morphism of projective varieties takes closed sets to closed sets."
In particular, let $X,Y$ be projective varieties of $\mathbb{P}^m,\mathbb{P}^n$ respectively and let $f: X \rightarrow Y$ be a morphism. If the graph $G_f = \left\{ (x,f(x)), x \in X \right\} \subset \mathbb{P}^m \times Y$ was a closed subset of $\mathbb{P}^m \times Y$, then statement (1) would readily apply and $f(X)$ would be closed in $Y$ as desired. However, why need $G_f$ be closed in $\mathbb{P}^m \times Y$?
Reference: See Theorem 1 and above in Akhil Mathew's post here: http://amathew.wordpress.com/2010/10/23/a-projective-morphism-is-proper/#comments
PS: Thanks to Mariano for a correction.