I am having trouble proving the following claim and would be glad if someone could help me out.
Claim: Let $\mathbb P$ denote n-dimensional projective space, and let $F$ be a coherent sheaf on $\mathbb P$. Then there exists some integer k such that $F\otimes O(k)$ is generated by a finite number of global sections.
I think I know how to begin: Let $\mathbb P^n=\bigcup U_i$ be the usual cover by open affines, and observe that $F\restriction U_i$ is finitely generated (as a coherent sheaf on an affine variety).
Now I would like to choose generators of all the $F\restriction U_i$ and tensor them with elements of $O(k)(U_i)$ (for some k) to obtain sections of $F\otimes O(k)$ which lift to global sections generating this sheaf.
The problem is that I don't understand well enough what sections of $F\restriction U_i$ look like, so I don't know what to tensor with.
Thanks in advance!