Globally generated line bundle without finite generators ($X={\coprod }_{m\ge 1}{\mathbb{P}}^m,\mathcal{L}{|}_{{\mathbb{P}}^m}=\mathcal{O}(1)$)

The displayed line bundle is globally generated, but no finite family of global sections generates it. A family of $N$ sections restricts to $N$ linear forms on ${\mathbb{P}}^m$; for $m\ge N$, these have a common zero. This shows why finite global generation on a quasi-compact scheme requires its finiteness hypothesis.