= Solution
For each point $x$, choose a <global section> $s$ whose <germ> $s_x$ generates the rank-one free $\mathcal O_{X,x}$-module $\mathcal L_x$. The section $s\otimes s$ is global, and <tensor products> commute with taking <stalks>. Therefore
$$
(s\otimes s)_x=s_x\otimes s_x
$$
generates $(\mathcal L\otimes_{\mathcal O_X}\mathcal L)_x$. Thus \b[the tensor square is globally generated]. More generally, the <tensor product of globally generated line bundles> is globally generated: choose a generator section for each factor at the point and tensor them. This works on every scheme, including nonreduced schemes.
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