Lemma \ref{lemma-ample-immersion-into-proj} is (\ref{item-ample}) $\Rightarrow$ (\ref{item-immersion}). Lemmas \ref{lemma-ample-power-ample} and \ref{lemma-open-in-proj-ample} provide the implication (\ref{item-ample}) $\Leftarrow$ (\ref{item-immersion}). The implications (\ref{item-immersion}) $\Rightarrow$ (\ref{item-s-affine-basis}) $\Rightarrow$ (\ref{item-s-basis}) are clear from Constructions, Section \ref{constructions-section-proj}. Lemma \ref{lemma-affine-s-opens} is (\ref{item-s-basis}) $\Rightarrow$ (\ref{item-ample}). Thus we see that the first 4 conditions are all equivalent. \medskip\noindent Assume the equivalent conditions (1) -- (4). Note that in particular $X$ is separated (as an open subscheme of the separated scheme $\text{Proj}(S)$). Let $\mathcal{F}$ be a quasi-coherent sheaf on $X$. Choose $s \in S_{+}$ homogeneous such that $X_s$ is affine. We claim that any section $m \in \Gamma(X_s, \mathcal{F})$ is in the image of one of the maps displayed in (\ref{item-qc-gg}) above. This will imply (\ref{item-qc-gg}) since these affines $X_s$ cover $X$. Namely, by Lemma \ref{lemma-invert-s-sections} we may write $m$ as the image of $m' \otimes s^{-n}$ for some $n \geq 1$, some $m' \in \Gamma(X, \mathcal{F} \otimes \mathcal{L}^{\otimes n})$. This proves the claim. \medskip\noindent Clearly (\ref{item-qc-gg}) $\Rightarrow$ (\ref{item-qc-i-gg}). Let us assume (\ref{item-qc-i-gg}) and prove $\mathcal{L}$ is ample. Pick $x \in X$. Let $U \subset X$ be an affine open which contains $x$. Set $Z = X \setminus U$. We may think of $Z$ as a reduced closed subscheme, see Schemes, Section \ref{schemes-section-reduced}. Let $\mathcal{I} \subset \mathcal{O}_X$ be the quasi-coherent sheaf of ideals corresponding to the closed subscheme $Z$. By assumption (\ref{item-qc-i-gg}), there exists an $n \geq 1$ and a section $s \in \Gamma(X, \mathcal{I} \otimes \mathcal{L}^{\otimes n})$ such that $s$ does not vanish at $x$ (more precisely such that $s \not \in \mathfrak m_x \mathcal{I}_x \otimes \mathcal{L}_x^{\otimes n}$). We may think of $s$ as a section of $\mathcal{L}^{\otimes n}$. Since it clearly vanishes along $Z$ we see that $X_s \subset U$. Hence $X_s$ is affine, see Lemma \ref{lemma-affine-cap-s-open}. This proves that $\mathcal{L}$ is ample. At this point we have proved that (1) -- (6) are equivalent. \medskip\noindent Assume the equivalent conditions (1) -- (6). In the following we will use the fact that the tensor product of two sheaves of modules which are globally generated is globally generated without further mention (see Modules, Lemma \ref{modules-lemma-tensor-product-globally-generated}). By (1) we can find elements $s_i \in S_{d_i}$ with $d_i \geq 1$ such that $X = \bigcup_{i = 1, \ldots, n} X_{s_i}$. Set $d = d_1\ldots d_n$. It follows that $\mathcal{L}^{\otimes d}$ is globally generated by $$ s_1^{d/d_1}, \ldots, s_n^{d/d_n}. $$ This means that if $\mathcal{L}^{\otimes j}$ is globally generated then so is $\mathcal{L}^{\otimes j + dn}$ for all $n \geq 0$. Fix a $j \in \{0, \ldots, d - 1\}$. For any point $x \in X$ there exists an $n \geq 1$ and a global section $s$ of $\mathcal{L}^{j + dn}$ which does not vanish at $x$, as follows from (\ref{item-qc-gg}) applied to $\mathcal{F} = \mathcal{L}^{\otimes j}$ and ample invertible sheaf $\mathcal{L}^{\otimes d}$. Since $X$ is quasi-compact there we may find a finite list of integers $n_i$ and global sections $s_i$ of $\mathcal{L}^{\otimes j + dn_i}$ which do not vanish at any point of $X$. Since $\mathcal{L}^{\otimes d}$ is globally generated this means that $\mathcal{L}^{\otimes j + dn}$ is globally generated where $n = \max\{n_i\}$. Since we proved this for every congruence class mod $d$ we conclude that there exists an $n_0 = n_0(\mathcal{L})$ such that $\mathcal{L}^{\otimes n}$ is globally generated for all $n \geq n_0$. At this point we see that if $\mathcal{F}$ is globally generated then so is $\mathcal{F} \otimes \mathcal{L}^{\otimes n}$ for all $n \geq n_0$. \medskip\noindent We continue to assume the equivalent conditions (1) -- (6). Let $\mathcal{F}$ be a quasi-coherent sheaf of $\mathcal{O}_X$-modules of finite type. Denote $\mathcal{F}_n \subset \mathcal{F}$ the image of the canonical map of (\ref{item-qc-gg}). By construction $\mathcal{F}_n \otimes \mathcal{L}^{\otimes n}$ is globally generated. By (\ref{item-qc-gg}) we see $\mathcal{F}$ is the sum of the subsheaves $\mathcal{F}_n$, $n \geq 1$. By Modules, Lemma \ref{modules-lemma-finite-type-quasi-compact-colimit} we see that $\mathcal{F} = \sum_{n = 1, \ldots, N} \mathcal{F}_n$ for some $N \geq 1$. It follows that $\mathcal{F} \otimes \mathcal{L}^{\otimes n}$ is globally generated whenever $n \geq N + n_0(\mathcal{L})$ with $n_0(\mathcal{L})$ as above. We conclude that (1) -- (6) implies (\ref{item-c-gg}). \medskip\noindent Assume (\ref{item-c-gg}). Let $\mathcal{F}$ be a quasi-coherent sheaf of $\mathcal{O}_X$-modules of finite type. By (\ref{item-c-gg}) there exists an integer $n \geq 1$ such that the canonical map $$ \Gamma(X, \mathcal{F} \otimes_{\mathcal{O}_X} \mathcal{L}^{\otimes n}) \otimes_{\mathbf{Z}} \mathcal{L}^{\otimes -n} \longrightarrow \mathcal{F} $$ is surjective. Let $I$ be the set of finite subsets of $\Gamma(X, \mathcal{F} \otimes_{\mathcal{O}_X} \mathcal{L}^{\otimes n})$ partially ordered by inclusion. Then $I$ is a directed partially ordered set. For $i = \{s_1, \ldots, s_{r(i)}\}$ let $\mathcal{F}_i \subset \mathcal{F}$ be the image of the map $$ \bigoplus\nolimits_{j = 1, \ldots, r(i)} \mathcal{L}^{\otimes -n} \longrightarrow \mathcal{F} $$ which is multiplication by $s_j$ on the $j$th factor. The surjectivity above implies that $\mathcal{F} = \colim_{i \in I} \mathcal{F}_i$. Hence Modules, Lemma \ref{modules-lemma-finite-type-quasi-compact-colimit} applies and we conclude that $\mathcal{F} = \mathcal{F}_i$ for some $i$. Hence we have proved (\ref{item-c-q}). In other words, (\ref{item-c-gg}) $\Rightarrow$ (\ref{item-c-q}). \medskip\noindent The implication (\ref{item-c-q}) $\Rightarrow$ (\ref{item-c-i-q}) is trivial. \medskip\noindent Finally, assume (\ref{item-c-i-q}). Let $\mathcal{I} \subset \mathcal{O}_X$ be a quasi-coherent sheaf of ideals. By Lemma \ref{lemma-quasi-coherent-colimit-finite-type} (this is where we use the condition that $X$ be quasi-separated) we see that $\mathcal{I} = \colim_\alpha I_\alpha$ with each $I_\alpha$ quasi-coherent of finite type. Since by assumption each of the $I_\alpha$ is a quotient of negative tensor powers of $\mathcal{L}$ we conclude the same for $\mathcal{I}$ (but of course without the finiteness or boundedness of the powers). Hence we conclude that (\ref{item-c-i-q}) implies (\ref{item-qc-i-gg}). This ends the proof of the proposition.