How do we know that Jesus fulfilled the prophecies in the Old Testament?

I have heard from Christians that Christianity is proven to be the correct religion because Jesus fulfilled over 300 prophecies written before his time in the Old Testament. How do Christians know that he actually fulfilled these prophecies? It seems to me that he may have fulfilled some of them, but any that he did not could have been added to accounts of his life later on when they were written down. How do Christians know that this is not the case?