To prove if time travel exists, you shouldn't go reading a Modern Physics book or Scientific American, I propose you could do it all but conclusively with a study of history.

If people have, in the future, figured out how to travel to the past, I assume there will be traces. Traces that further travel is, for some reason, unable to erase or prevent. I have no way to know or prove this assumption is true. If, however, it is, we should be able to find these traces in history.

We are looking for someone who appears suddenly in a culture they are unfamiliar with, and would have to use their advanced knowledge of either technology/science or knowledge of future events to survive. Look for people who had advanced ideas or are generally viewed as "ahead of their time," whose lifestyle or views dramatically clashed with their culture (they had difficulty adapting to what they see as a primitive or stupid cultural rules like slavery, inequality, religous rule, etc), and/or whose origins and birthpalce are in question. Someone like say Galileo or better yet Bohr. Detailed examinations of their works could share clues to higher knowledge not possible to the time period (ie a^x+b^y=c^z with no proof in the margin of a math text). Further, look for isolated discoveries (use of geometry before greeks, etc). While this could be simply derived works (greeks learned geometry from someone else) geographically isolated areas help support this (geometry in South America). An examine of how unique this knowledge is to see the chance of it independentally being discovered (ie the pyramids look the same in SE asia, Egypt, and SA not because of aliens, but because the only way to build tall structures with primitive materials is to distribute the weight of the higher levels over larger, lower layers, [ie a triangle], and hence a pyramid)

As someone could influence events without seeming personally responsible, examine people in an advising capacity, especially during war periods. Looking at events that all seemed to "by chance" favor one side, and apply a normal distribution to these outcomes. Did the outcomes favor one side more than they should have. Then see if technology/intrigue unnaturally influenced the distribution (ie did the US always seem to have luck on their side because they had borken codes?).

It seems to me the ideal time frame to examine would be 500 AD - 1950. During this time the ability/money/culture to educate large quantities of people were minimized, and an outstanding person would be easier to find.

Discuss, and remind me never to eat lots of chicken while staying up until 3am readin "Rise and Fall of the 3rd Reich"