Here's a thought experiment to prove that time travel is impossible*:
Suppose you invented time travel right now, today. Now let's say you're a clever, sneaky person, and you managed to use it to make yourself wealthy and save some people you care about from accidental deaths or by getting them preventative care for illnesses that otherwise would have killed them, all while keeping it completely secret. You destroy your time travel device once you're satisfied with the benefit you've taken from it, and you never reveal it to anyone.
Doesn't matter. Because if a thing is possible, someone will discover it again. And again. And again if necessary. In 10 years or 10,000 years someone is going to discover it and they're going to fail to be as clever and sneaky as you, and word is going to get out. Some government or private group may control it for a while, but eventually (remember, we have all of time to work with here), the tech WILL escape into the wild, and when it does, there's going to be all sorts of shenanigans. Time theft (stealing cool stuff from the past and future), and more importantly for our purposes, temporal tourism.
Now, to begin with it may be limited to the wealthy who will go to see the signing of the Magna Carta and such with guides who will dress them in period attire and help them to be very careful not to disrupt the timeline. But eventually, Joe Schlub and his family are going to want to take a vacation back to before it was necessary to wear breathing masks to survive outside on Earth, and the best place for the family budget is going to be in some cheesy hotel in your town in 2014, because all of the best spots will have been taken by other temporal tourists.
So there you go. The fact that you are not, as we speak, surrounded by temporal tourists, while being interviewed by future historians about what it was like to live in 2014, proves that time travel is not possible.
*There's actually two other things that this argument could prove:
1. Time travel involves branching of timelines, and the reason we've never encountered a solid, provable temporal anomaly is that WE are the ORIGINAL branch, so there are no changes here. All the fun stuff happens elsewhere.
2. Every civilization throughout space and time wipes itself out before reaching a level of tech that allows time travel.
But both of those are so depressing that I'd really just prefer to think that time travel is impossible.