Climbing up a filtration of a sheaf to show noetherian vanishing cohomology implies affine

Question

Let $(X, \mathcal{O}_{X})$ be a noetherian scheme. I am trying to show that if $H^{1}(X, \mathscr{I})$ vanishes for all coherent sheaves of ideals $\mathscr{I}$, then $X$ is affine. I am following the proof in Hartshorne III 3.7. This seems to be the standard proof. I am also supplementing it with the notes here (page 9-10). So I am comfortable with everything up until a point. Say we have constructed the sequence $$ 0 \longrightarrow \mathcal{F} \longrightarrow \mathcal{O}_{X}^{r} \longrightarrow \mathcal{O}_{X} \longrightarrow 0. $$ I am also comfortable with the filtration, $$ \mathcal{F} = \mathcal{F} \cap \mathcal{O}_{X}^{r} \supseteq \mathcal{F} \cap \mathcal{O}_{X}^{r-1} \supseteq \mathcal{F} \cap \mathcal{O}_{X}^{r-2} \supseteq \cdots \supseteq \mathcal{F} \cap \mathcal{O}_{X} \supseteq 0. $$ Label $\mathcal{F}_{i} := \mathcal{F} \cap \mathcal{O}_{X}^{r-i}$. Then it is easy enough to deduce that $\mathcal{F}_{i} / \mathcal{F}_{i+1}$ is a coherent sheaf of ideals. However, it is then claimed that we can "climb up the filtration" using the long exact sequence for cohomology to deduce that $H^{1}(X, \mathcal{F})=0$. It is not at all clear to me how to do this, or which long exact sequence we want to take. Say we consider, $$ 0 \longrightarrow \mathcal{F}_{r-1} \longrightarrow \mathcal{F}_{r-2} \longrightarrow \mathcal{F}_{r-2} / \mathcal{F}_{r-1} \longrightarrow 0. $$ The long exact sequence would then read $$ \cdots\rightarrow \Gamma(X, \mathcal{F}_{r-2}/\mathcal{F}_{r-1}) \rightarrow H^{1}(X, \mathcal{F}_{r-1}) \simeq 0 \rightarrow H^{1}(X, \mathcal{F}_{r-2}) \rightarrow H^{1}(X, \mathcal{F}_{r-2}/\mathcal{F}_{r-1}) \simeq 0 \rightarrow H^{2}(X, \mathcal{F}_{r-1}) \rightarrow \cdots $$ Where we have used the assumption that the first cohomology vanishes for coherent sheaves of ideals. But I fail to see how we can make any inductive step here to get $H^{1}(X, \mathcal{F}_{r-2})$ to vanish. It is sandwiched between two $0$ terms, but unless we can get an extra $0$ term on one side that doesn't tell us much. Am I just taking the wrong short exact sequence to get the corresponding long exact sequence? Or am I missing something more obvious?