Assume $K \in D(\mathcal{A})$ is a compact object. Represent $K$ by a complex $K^\bullet$ and consider the map $$ K^\bullet \longrightarrow \bigoplus\nolimits_{n \geq 0} \tau_{\geq n} K^\bullet $$ where we have used the canonical truncations, see Homology, Section \ref{homology-section-truncations}. This makes sense as in each degree the direct sum on the right is finite. By assumption this map factors through a finite direct sum. We conclude that $K \to \tau_{\geq n} K$ is zero for at least one $n$, i.e., $K$ is in $D^{-}(R)$. \medskip\noindent We may represent $K$ by a bounded above complex $K^\bullet$ each of whose terms is a direct sum of objects from $S$, see Derived Categories, Lemma \ref{derived-lemma-subcategory-left-resolution}. Note that we have $$ K^\bullet = \bigcup\nolimits_{n \leq 0} \sigma_{\geq n}K^\bullet $$ where we have used the stupid truncations, see Homology, Section \ref{homology-section-truncations}. Hence by Derived Categories, Lemmas \ref{derived-lemma-colim-hocolim} and \ref{derived-lemma-commutes-with-countable-sums} we see that $1 : K^\bullet \to K^\bullet$ factors through $\sigma_{\geq n}K^\bullet \to K^\bullet$ in $D(R)$. Thus we see that $1 : K^\bullet \to K^\bullet$ factors as $$ K^\bullet \xrightarrow{\varphi} L^\bullet \xrightarrow{\psi} K^\bullet $$ in $D(\mathcal{A})$ for some complex $L^\bullet$ which is bounded and whose terms are direct sums of elements of $S$. Say $L^i$ is zero for $i \not \in [a, b]$. Let $c$ be the largest integer $\leq b + 1$ such that $L^i$ a finite direct sum of elements of $S$ for $i < c$. Claim: if $c < b + 1$, then we can modify $L^\bullet$ to increase $c$. By induction this claim will show we have a factorization of $1_K$ as $$ K \xrightarrow{\varphi} L \xrightarrow{\psi} K $$ in $D(\mathcal{A})$ where $L$ can be represented by a finite complex of finite direct sums of elements of $S$. Note that $e = \varphi \circ \psi \in \text{End}_{D(\mathcal{A})}(L)$ is an idempotent. By Derived Categories, Lemma \ref{derived-lemma-projectors-have-images-triangulated} we see that $L = \Ker(e) \oplus \Ker(1 - e)$. The map $\varphi : K \to L$ induces an isomorphism with $\Ker(1 - e)$ in $D(R)$ and we conclude. \medskip\noindent Proof of the claim. Write $L^c = \bigoplus_{\lambda \in \Lambda} E_\lambda$. Since $L^{c - 1}$ is a finite direct sum of elements of $S$ we can by assumption (2) find a finite subset $\Lambda' \subset \Lambda$ such that $L^{c - 1} \to L^c$ factors through $\bigoplus_{\lambda \in \Lambda'} E_\lambda \subset L^c$. Consider the map of complexes $$ \pi : L^\bullet \longrightarrow (\bigoplus\nolimits_{\lambda \in \Lambda \setminus \Lambda'} E_\lambda)[-i] $$ given by the projection onto the factors corresponding to $\Lambda \setminus \Lambda'$ in degree $i$. By our assumption on $K$ we see that, after possibly replacing $\Lambda'$ by a larger finite subset, we may assume that $\pi \circ \varphi = 0$ in $D(\mathcal{A})$. Let $(L')^\bullet \subset L^\bullet$ be the kernel of $\pi$. Since $\pi$ is surjective we get a short exact sequence of complexes, which gives a distinguished triangle in $D(\mathcal{A})$ (see Derived Categories, Lemma \ref{derived-lemma-derived-canonical-delta-functor}). Since $\Hom_{D(\mathcal{A})}(K, -)$ is homological (see Derived Categories, Lemma \ref{derived-lemma-representable-homological}) and $\pi \circ \varphi = 0$, we can find a morphism $\varphi' : K^\bullet \to (L')^\bullet$ in $D(\mathcal{A})$ whose composition with $(L')^\bullet \to L^\bullet$ gives $\varphi$. Setting $\psi'$ equal to the composition of $\psi$ with $(L')^\bullet \to L^\bullet$ we obtain a new factorization. Since $(L')^\bullet$ agrees with $L^\bullet$ except in degree $c$ and since $(L')^c = \bigoplus_{\lambda \in \Lambda'} E_\lambda$ the claim is proved.