Consider $X$ a projective variety with $\operatorname{Pic}X\cong \mathbb Z$. So $\mathcal L\cong \mathcal O_X(m)$ for any invertible sheaf $\mathcal L$ and for some $m\in \mathbb Z$. I want to show that either $\mathcal O_X(m)$, $m>0$ have nonzero global sections or $\mathcal O_X(m)$, $m<0$ have nonzero global sections.
If $\mathcal L,\mathcal L^{-1}$ have nonzero global sections then $\mathcal L\cong \mathcal O_X$ so only one of $\mathcal O_X(-m),\mathcal O_X(m)$ has global sections ($m\neq 0$). But a priori there is no reason that if there are sheaves with nonzero global sections, they correspond to positive or negative integers. One could have that both $\mathcal O_X(-2),\mathcal O_X(3)$ have nonzero global sections, alternating the sign.
Since $X$ is projective we have a closed immersion $\iota\colon X\hookrightarrow \mathbb P^n_K$, so $\iota^*\mathcal O(1)$ is very ample and has global sections. So there exists some $k\in\mathbb Z$ such that $\mathcal O_X(k)$ has nonzero global section but it does not help much.
I guess showing that either $\mathcal O_X(1)$ or $\mathcal O_X(-1)$ has nonzero global sections would be enough. Indeed if $\mathcal O_X(1)$ has global sections, then any power $\mathcal O_X(m)$ has global sections, same for $\mathcal O_X(-1)$. But I don't know this is true. Any help ?