Achilles definitely catches the tortoise.

The way the story is told, they keep describing snapshots in time that are closer and closer to the time the tortoise is caught without telling you about that moment.

Like if he catches him at noon and you tell a story describing 11:59, then 11:59:30, then 11:59:45, then 11:59:52.5, etc.

You can certainly tell a story like that where you never get to noon. But we all know how time works. Eventually noon will come.