The Stacks project

Proof. There are at most $\max \{ |R|, \aleph _0\} $ affine opens of $\mathop{\mathrm{Spec}}(R)$. This is clear since any affine open $U \subset \mathop{\mathrm{Spec}}(R)$ is a finite union of principal opens $D(f_1) \cup \ldots \cup D(f_ n)$ and hence the number of affine opens is at most $\sup _ n |R|^ n = \max \{ |R|, \aleph _0\} $, see [Ch. I, 10.13, Kunen]. On the other hand, we see that $\Gamma (U, \mathcal{O}) \subset R_{f_1} \times \ldots \times R_{f_ n}$ and hence $|\Gamma (U, \mathcal{O})| \leq \max \{ \aleph _0, |R_{f_1}|, \ldots , |R_{f_ n}|\} $. Thus it suffices to prove that $|R_ f| \leq \max \{ \aleph _0, |R|\} $ which is omitted. $\square$