We study the properties of the free boundaries and the corresponding hitting times in the context of optimal stopping in discrete time. We first prove the continuity of the map from the boundaries to the expected value of the corresponding stopping policy both in the supremum norm and also in a weaker, novel topology induced by the relaxed $L^\infty$ metric that we introduce. The latter is particularly useful when the optimal stopping boundary is only proved to be semicontinuous. Secondly, we study the connection between the hitting times, and their relaxations as widely employed in recent numerical methods. All these results together with the universal approximation capability of neural networks and the notion of inf/sup convolution are then used to provide a convergence analysis for the algorithm in [Reppen, Soner, and Tissot-Daguette, Neural Optimal Stopping Boundary, 2025] for the numerical resolution of the exercise regions arising in the analysis of Bermudan type option.

22 pages, 5 figures