I think this is an excellent question. I believe all belief is subjective. You can't prove divine inspiration the way you would prove 1+1=2. It depends on how compelling you find the evidence available to you, which really varies from person to person. For those people who lived with Muhammad, several were made to believe in him due to his personality. Some believed in him almost immediately, after only a brief encounter. By all accounts - even by his most vehement critics - Muhammad was really an extraordinary man. But for the rest of us who have never met him, and only know about his life through second person accounts, why do we believe?
I think the most compelling case in support of Muhammad's claim to divine inspiration is the Qur'an - that was his miracle. Perhaps, there's no other book that has affected the development of a language to the extent the Qur'an had on Arabic. This fact is even more extraordinary considering that the Prophet had never been a poet or known to have studied or written any book prior to his claim of being divinely inspired:
"And you did not recite before it any book, nor did you transcribe one
with your right hand, for then could those who say untrue things have
About the Qur'an, Muhammad himself said,
"Every Prophet was given miracles because of which people believed,
but what I have been given, is Divine Inspiration which Allah has
revealed to me. So I hope that my followers will outnumber the
followers of the other Prophets on the Day of Resurrection."
All this doesn't prove anything of course. It's for the individual to decide if Muhammad's quite extraordinary life (even by secular accounts) and the unlikeliness that he authored the Qur'an (given his non-literate background), is enough for him to accept Muhammad as truly divinely inspired.
Although the Rabbi seemed to hold a Moses was right, Jesus and Muhammad were both wrong point of view, Muslims believe Moses, Jesus and Muhammad to be all divinely inspired, although they received their inspirations differently. So with Muslims, it's not Muhammad was right, Jesus and Moses were both wrong. This wasn't your question, but I thought it was important to point this out.