The question $\mathcal{L}$ is very ample, $\mathcal{U}$ is generated by global sections $\Rightarrow$ $\mathcal{L} \otimes \mathcal{U}$ is very ample asks for a proof of the following statement:
Let $X$ be Noetherian and of finite type over a Noetherian ring, $\mathcal L$ and $\mathcal M$ line bundles on $X$ with $\mathcal L$ very ample and $\mathcal M$ generated by global sections. Then $\mathcal L \otimes \mathcal M$ is very ample.
I followed the proof in the comments there but failed to find a step where the finite type assumption is necessary. On the other hand, this is exercise III.7.5 (d) in Hartshorne's Algebraic Geometry and there this assumption is stated explicitly for part (d).
So is this assumption necessary? (If no, does the proof in the linked thread use it somewhere nonetheless and I didn't realise?)