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For a unital spectral triple $(\mathcal{A}, H,D)$, we study when itstruncation converges to itself. The spectral truncation is obtained by usingthe spectral projection $P_{\Lambda}$ of $D$ onto $[-\Lambda,\Lambda]$ to dealwith the case where only a finite range of energy levels of a physical systemis available. By restricting operators in $\mathcal{A}$ and $D$ to$P_{\Lambda}H$, we obtain a sequence of operator system spectral triples$\{(P_{\Lambda}\mathcal{A}P_{\Lambda},P_{\Lambda}H,P_{\Lambda}DP_{\Lambda})\}_{\Lambda}$.We prove that if the spectral triple is the one constructed using a discretegroup with polynomial growth, then the sequence of operator systems$\{P_{\Lambda}\mathcal{A}P_{\Lambda}\}_{\Lambda}$ converges to $\mathcal{A}$ inthe sense of quantum Gromov-Hausdorff convergence with respect to the Lip-normcoming from high order derivatives.

Quantum Gromov-Hausdorff convergence of spectral truncations for groups with polynomial growth