Did Sauron Really Know Frodo?

[Midgardsormen]

Beren, why should he put 'two and two togehter' as you say?? Who in their right minds would enter Mordor voluntarily?? That was the whole point! Boromir said that it was folly, and even Gandalf conceeded that the chances of success were extremely small, but that Frodo and Sam had the best chance because it was totally unbelievable that two little helpless halflings would walk INTO Mordor with the One Ring!! He knew some Hobbits had it, but assumed that they would go to Minas Tirith with it, and when Sauron saw Aragorn in the Palantir, he was convinced that Aragorn had the ring even if he was not wearing it, and would try to wrest it from Sauron when the armies met. Even when Frodo got captured, although they stripped him to make sure, the Orcs (and Sauron....who knew about him once he got the Mithril shirt) thought he (and his 'Elven' companion) were sent to spy on Sauron.
So I think Gandalf was being wise....and to further help them they marched to the Morannen, although he knew that they might die before they even knew whether Frodo was successful or not. (and when they saw the Mithril shirt etc. they assumed all was lost but they still fought!!)
So, although Sauron was aware of a halfling having the ring, he could in no way concieve of it not being used against him.