Here's why:
* He doesn't know about the diadem until the end of the book: Harry only learns about the diadem's existence in the Room of Requirement when he and his friends are trying to find Voldemort's last Horcrux.
* He doesn't know about the diadem being a Horcrux until he finds it: He doesn't discover it was used by Voldemort until he finds the diadem in the Room of Requirement. He doesn't know about the diadem's hidden Horcrux until he finds the diadem.
* He knows there are six Horcruxes and he's found five: Prior to finding the diadem, Harry is aware that there are six Horcruxes and he has found five (Tom Riddle's diary, Marvolo Gaunt's ring, Helga Hufflepuff's cup, Salazar Slytherin's locket, and Nagini).
It's not Harry who believes the diadem is the last Horcrux, but Voldemort. He tells Harry that the diadem is the last Horcrux he made.
However, this proves to be a lie. The final Horcrux is actually Nagini. This is revealed when Harry kills her in the battle of Hogwarts.
So, while Harry discovers the diadem in his search for the final Horcrux, he doesn't believe it is the final Horcrux himself.