I'm browsing through my notes in algebraic geometry and I'm struggling with a theorem.
So recall that in the Zariski topology in the affine space $\mathbb A^n$ the closed sets are zero loci of polynomials in $k[x_1, x_2,\ldots, x_n]$.
Now consider the projective space $\mathbb P^n$ and its standard cover with affine open sets $\mathbb P^n=U_0\cup U_1\cup \ldots \cup U_n$, where $U_i=\{[x_0, x_2,\ldots, x_n], x_i\ne 0\}$.
Now define the Zariski topology on $\mathbb P^n$ to be the glueing topology, i.e. $X\subset \mathbb P^n$ is closed iff each $X\cap U_i$ is Zariski closed in $U_i$ as an affine set. I want to show that the closed sets in $\mathbb P^n$ are precisely the zero loci of homogeneous polynomials.
While it is easy to show that zero loci of homogeneous polynomials are closed sets in this sense, I cannot show that the converse is also true and I would appreciate some help.