Proof.
The dg nerve of the dg category of (unbounded) chain complexes is an \(\infty\)-category by [Lur17, Prop. 1.3.1.10] and stable by [Lur17, Prop. 1.3.2.10]. The full dg subcategory of bounded chain complexes is closed under shifts and formation of mapping cones, and thus its dg nerve \({\mathbf K}^b(\mathcal A)\) is itself a stable \(\infty\)-category, by [Lur17, Lem. 1.1.3.3] and the discussion after [Lur17, Proof of Prop. 1.3.2.10]. ◻