One of the most interesting questions in computer science is of course whether $P = NP$ or $P \neq NP$. If one wants to prove that $P \neq NP$ one can try to prove that an NPC problem is not solvable in polynomial time. However, I can't quite figure out how it would be possible to prove such a thing.
How can you prove that there is no better solution possible to a problem than a solution which has a certain time complexity?
I understand that there are no problems known in NP for which is proven that it cannot be solved in polynomial time, but are there problems known for which a proof does exists that it cannot be solved in for example constant or linear time? Perhaps an example of such a problem can help me understand this better.
EDIT: If you down vote this question could you please at least post a comment why you did so? I'd love to improve my question, but down votes alone don't tell me what I did wrong.