As you've brought up time travel... you can use it to prove that either the life span of our universe is finite, (it will end at some point) or time travel will never ever exist...
If the universe were infinite AND time travel is possible then at some point it will be created/invented and with an infinite future to come back from our current time should be totally over-run by time travellers, which it isn't.
Therefore either, the universe is finite, or time travel is impossible, or both.