Gromov-Hausdorff limit of the p-Adic integers

The p-adic numbers described by Kurt Hensel in 1897 are still quite mysterious algebraic structures that need to be studied in detail. For a prime number p, p-adic numbers Q_p can be obtained as the completion of rational numbers according to the p-absolute value norm. In addition, p-adic integers Z_p can be briefly expressed as the unit disk in Q_p. On the other hand, Gromov-Hausdorff distance is one of the most important tools in the field of metric geometry to measure the distance between two different (compact) metric spaces. The Gromov-Hausdorff distance between two metric spaces can be defined as the infimum of the Hausdorff distances between possible isometric embeddings of them into a third metric space. In this work, it is studied with the problem of whether the sequence of compact sets Z_p converges to a metric space in the sense of Gromov-Hausdorff. In the sense of classical Gromov-Housdorff convergence, it is shown that p-adic integers cannot converge to a metric space. Inspired by notion of the convergence of a sequence of pointed metric spaces, a new convergence definition, which is considered to be more significant for the convergence of such metric spaces, is given and shown that the sequence of p-adic integers converges to a set of non-negative integers equipped with discrete metric. Keywords: Gromov-Hausdorff distance, Absolute value function, p-adic numbers, p-adic integers, Ostrowski Theorem