 We prove that these barcodes are stable under perturbations in the input DMS. In order to formalize the notion of perturbation we introduce a suitable distance between DMSs and we then prove that the value of this distance between any two DMSs admits as a lower bound the bottleneck distance between the Rips barcodes associated to each of two input DMSs.
<meaning-changed>  We prove that these barcodes are stable under perturbations in the input DMS. In order to formalize the notion of perturbation we introduce a suitable distance between DMSs and we then prove that the value of this distance between any two DMSs admits as a lower bound the bottleneck distance between the Rips barcodes associated to each of two input DMSs.
 We prove that these barcodes are stable under perturbations in the input DG under a suitable distance between DMSs and we then prove that the value of this distance between any two DMSs admits as a lower bound the bottleneck distance between the Rips barcodes associated to each of two input DMSs.