It is immediate from the definition that (1) implies (2) and (2) implies (3). It is clear that (5) implies (4). \medskip\noindent Assume (3) holds for the affine open covering $S = \bigcup V_i$. We are going to show (5) holds. Since each $f^{-1}(V_i)$ has an ample invertible sheaf we see that $f^{-1}(V_i)$ is separated (Properties, Lemma \ref{properties-lemma-ample-separated}). Hence $f$ is separated. By Schemes, Lemma \ref{schemes-lemma-push-forward-quasi-coherent} we see that $\mathcal{A} = f_*(\bigoplus_{d \geq 0} \mathcal{L}^{\otimes d})$ is a quasi-coherent graded $\mathcal{O}_S$-algebra. Denote $\psi : f^*\mathcal{A} \to \bigoplus_{d \geq 0} \mathcal{L}^{\otimes d}$ the adjunction mapping. The description of the open $U(\psi)$ in Constructions, Section \ref{constructions-section-invertible-relative-proj} and the definition of ampleness of $\mathcal{L}|_{f^{-1}(V_i)}$ show that $U(\psi) = X$. Moreover, Constructions, Lemma \ref{constructions-lemma-invertible-map-into-relative-proj} part (3) shows that the restriction of $r_{\mathcal{L}, \psi}$ to $f^{-1}(V_i)$ is the same as the morphism from Properties, Lemma \ref{properties-lemma-map-into-proj} which is an open immersion according to Properties, Lemma \ref{properties-lemma-ample-immersion-into-proj}. Hence (5) holds. \medskip\noindent Let us show that (4) implies (1). Assume (4). Denote $\pi : \underline{\text{Proj}}_S(\mathcal{A}) \to S$ the structure morphism. Choose $V \subset S$ affine open. By Constructions, Definition \ref{constructions-definition-relative-proj} we see that $\pi^{-1}(V) \subset \underline{\text{Proj}}_S(\mathcal{A})$ is equal to $\text{Proj}(A)$ where $A = \mathcal{A}(V)$ as a graded ring. Hence $r_{\mathcal{L}, \psi}$ maps $f^{-1}(V)$ isomorphically onto a quasi-compact open of $\text{Proj}(A)$. Moreover, $\mathcal{L}^{\otimes d}$ is isomorphic to the pullback of $\mathcal{O}_{\text{Proj}(A)}(d)$ for some $d \geq 1$. (See part (3) of Constructions, Lemma \ref{constructions-lemma-invertible-map-into-relative-proj} and the final statement of Constructions, Lemma \ref{constructions-lemma-invertible-map-into-proj}.) This implies that $\mathcal{L}|_{f^{-1}(V)}$ is ample by Properties, Lemmas \ref{properties-lemma-open-in-proj-ample} and \ref{properties-lemma-ample-power-ample}. \medskip\noindent Assume (6). By the equivalence of (1) - (5) above we see that the property of being relatively ample on $X/S$ is local on $S$. Hence we may assume that $S$ is affine, and we have to show that $\mathcal{L}$ is ample on $X$. In this case the morphism $r_{\mathcal{L}, \psi}$ is identified with the morphism, also denoted $r_{\mathcal{L}, \psi} : X \to \text{Proj}(A)$ associated to the map $\psi : A = \mathcal{A}(V) \to \Gamma_*(X, \mathcal{L})$. (See references above.) As above we also see that $\mathcal{L}^{\otimes d}$ is the pullback of the sheaf $\mathcal{O}_{\text{Proj}(A)}(d)$ for some $d \geq 1$. Moreover, since $X$ is quasi-compact we see that $X$ gets identified with a closed subscheme of a quasi-compact open subscheme $Y \subset \text{Proj}(A)$. By Constructions, Lemma \ref{constructions-lemma-ample-on-proj} (see also Properties, Lemma \ref{properties-lemma-open-in-proj-ample}) we see that $\mathcal{O}_Y(d')$ is an ample invertible sheaf on $Y$ for some $d' \geq 1$. Since the restriction of an ample sheaf to a closed subscheme is ample, see Properties, Lemma \ref{properties-lemma-ample-on-closed} we conclude that the pullback of $\mathcal{O}_Y(d')$ is ample. Combining these results with Properties, Lemma \ref{properties-lemma-ample-power-ample} we conclude that $\mathcal{L}$ is ample as desired.