Let $a, b \in \mathbf{Z}$ be integers such that $\mathcal{F}^\bullet$ has tor amplitude in $[a, b]$ and such that $\mathcal{F}^n = 0$ for $n < a$. The existence of such a pair of integers follows from Cohomology, Lemma \ref{cohomology-lemma-perfect} and the fact that $X$ is quasi-compact. If $b < a$, then $\mathcal{F}^\bullet$ is zero in the derived category and the lemma holds. We will prove by induction on $b - a \geq 0$ that there exists a complex $\mathcal{E}^a \to \ldots \to \mathcal{E}^b$ with $\mathcal{E}^i$ finite locally free and a quasi-isomorphism $\mathcal{E}^\bullet \to \mathcal{F}^\bullet$. \medskip\noindent The base case is the case $b - a = 0$. In this case $H^b(\mathcal{F}^\bullet) = H^a(\mathcal{F}^\bullet) = \Ker(\mathcal{F}^a \to \mathcal{F}^{a + 1})$ is finite locally free. Namely, it is a finitely presented $\mathcal{O}_X$-module of tor dimension $0$ and hence finite locally free. See Cohomology, Lemmas \ref{cohomology-lemma-perfect} and \ref{cohomology-lemma-finite-cohomology} and Properties, Lemma \ref{properties-lemma-finite-locally-free}. Thus we can take $\mathcal{E}^\bullet$ to be $H^b(\mathcal{F}^\bullet)$ sitting in degree $b$. The rest of the proof is dedicated to the induction step. \medskip\noindent Assume $b > a$. Observe that $$ H^b(\mathcal{F}^\bullet) = \Ker(\mathcal{F}^b \to \mathcal{F}^{b + 1})/ \Im(\mathcal{F}^{b - 1} \to \mathcal{F}^b) $$ is a finite type quasi-coherent $\mathcal{O}_X$-module, see Cohomology, Lemmas \ref{cohomology-lemma-perfect} and \ref{cohomology-lemma-finite-cohomology}. Then we can find a coherent $\mathcal{O}_X$-module $\mathcal{F}$ and a map $$ \mathcal{F} \longrightarrow \Ker(\mathcal{F}^b \to \mathcal{F}^{b + 1}) $$ such that the composition with the projection onto $H^b(\mathcal{F}^\bullet)$ is surjective. Namely, we can write $\Ker(\mathcal{F}^b \to \mathcal{F}^{b + 1})$ as the filtered union of its coherent submodules by Properties, Lemma \ref{properties-lemma-quasi-coherent-colimit-finite-type} and then one of these will do the job. Next, we choose a finite locally free $\mathcal{O}_X$-module $\mathcal{E}^b$ and a surjection $\mathcal{E}^b \to \mathcal{F}$ using the resolution property of $X$. Consider the map of complexes $$ \alpha : \mathcal{E}^b[-b] \to \mathcal{F}^\bullet $$ and its cone $C(\alpha)^\bullet$, see Derived Categories, Definition \ref{derived-definition-cone}. We observe that $C(\alpha)^\bullet$ is nonzero only in degrees $\geq a$, has tor amplitude in $[a, b]$ by Cohomology, Lemma \ref{cohomology-lemma-cone-tor-amplitude}, and has $H^b(C(\alpha)^\bullet) = 0$ by construction. Thus we actually find that $C(\alpha)^\bullet$ has tor amplitude in $[a, b - 1]$. Hence the induction hypothesis applies to $C(\alpha)^\bullet$ and we find a map of complexes $$ (\mathcal{E}^a \to \ldots \to \mathcal{E}^{b - 1}) \longrightarrow C(\alpha)^\bullet $$ with properties as stated in the induction hypothesis. Unwinding the definition of the cone this gives a commutative diagram $$ \xymatrix{ \ldots \ar[r] & \mathcal{E}^{b - 2} \ar[r] \ar[d] & \mathcal{E}^{b - 1} \ar[r] \ar[d] & 0 \ar[r] \ar[d] & \ldots \\ \ldots \ar[r] & \mathcal{F}^{b - 2} \ar[r] & \mathcal{F}^{b - 1} \oplus \mathcal{E}^b \ar[r] & \mathcal{F}^b \ar[r] & \ldots } $$ It is clear that we obtain a map of complexes $(\mathcal{E}^a \to \ldots \to \mathcal{E}^b) \to \mathcal{F}^\bullet$. We omit the verification that this map is a quasi-isomorphism.