Your geometric series is not a summation of the steps or positions, but rather the time required to complete each step. Therefore your example is characterized by an identical geometric series to the model I used in my previous comment.

More generally, Zeno’s paradox can be succinctly resolved by citing the monotone convergence theorem. Every bounded, monotonically decreasing function converges. The time required to complete the infinite series of half steps converges, because (again, with the definition of a metric) the time required to complete each individual step decreases commensurate with the change in distance.