We have numerous religious textual references, and ideas to backup the idea of who is the Messiah; yet as we see with Yeshua's fulfilment of prophecy, some people will go against it through a lack of study...
So is it ever really possible to actually prove it for sure using careful exegesis or are people to argumentative to really listen?
In my opinion.
The first way one could prove to me that one might be the Messiah would be to stop trying to convince people that s/he is the Messiah.
If one was really the Messiah, I would think s/he would gain followers through his/her actions, not by saying, "I'm am the Messiah! Follow me and listen to what I say!!"
Of course, a miracle or two along with a divine healing would help convince me as well. But then again, those are actions, not words.