Chapter 1, Exercise 2.5: Properties of $\mathbb P^n$

(a) $\mathbb P^n$ is an noetherian topological space.

(b) Every algebraic set in $\mathbb P^n$ can be written uniquely as a finite union of irreducible algebraic sets, no one containing another. These are called its irreducible components.

This exercise is essentially identical to the material covered in Section 1, with no technical difficulties arising from passing to projective space.

(a) If $Y_1\supseteq Y_2\supseteq\dots$ is a descending chain in $\mathbb P^n$, taking ideals gives an ascending chain $I(Y_1)\subseteq I(Y_2)\subseteq\dots$ in $S$, but $S$ is noetherian, hence this stabilizes, hence so does the original chain.

(b) Proposition 1.5 works in general even if $X=\mathbb P^n$ (by (a)), and we are done.