How do Jews prove that Judaism is the truth?
1) We don't.
(as has been mentioned)
2) I; me; personally; not inclusive of anyone else; IMHO - also don't, but.....
G-d is the G-d of History. And, I love History. I specialize in loving Religious History. And, I really love Jewish; Christian; Muslim religious History. And....
Jewish History fascinates me....
Jews are a-Historical. Jews exist outside the ordinary paradigms of History.
In this year of somebody's lord, 2015, the entire planet Earth is governed by concepts that originated in the Jewish Torah.
Jews have always been insignificant in numbers and hugely influential in every aspect of life.
Jews have always been persecuted to the point of destruction and have always remained hugely influential on every aspect of life in This World.
Jews are the only "People" who have existed for over three thousand years in obscurity and neglect that have remained a hugely influential "People."
Statistically speaking, today, Jews do not even exist and yet - some organizations such as the UN, still believe that they are the root of all the world's problems.
The whole Kingdoms of Israel 2,000 years ago and before and the reestablishment of the Secular Jewish State of Israel thing. (This new Secular Jewish State of Israel, incidentally, fully expected, with perfectly logical reasoning, that "Judaism;" ie: "Orthodox Jews" would disappear by the mid 20th Century. Instead, "Orthodox Judaism" is growing by leaps and bounds transforming Israel and the Jewish world.)
There's a lot more but - this is all History.
Therefore, for me; as for me; "when I goooo.... I'm goin' like Elsie! Start by admitting from cradle to tomb isn't that long a stay!...." oh... sorry... channeling the wrong a-Historical Jewish influence.... (Cabaret)
Therefore, as for me - G-d is the G-d of History and He gave His Torah to the Children of Israel on Mt. Sinai.