LCKurtz said:Think about a polygonal approximation to the upper half of a circle.
davitykale said:Sorry, I'm really confused :/
davitykale said:How is f not Lipschitz if f_n(x) is? Doesn't f_n --> f where f is the semicircle?
davitykale said:Maybe I'm just confused...f is the semicircle, correct?
Yes, one example is the statement "There is no highest prime number." This statement is impossible to prove because even if we search for the highest prime number, there could always be a larger prime number that we haven't found yet.
No, there is no way to definitively prove that something is impossible. We can only provide evidence or logical reasoning to support the idea that something is impossible, but we cannot prove it with absolute certainty.
We can determine that something is impossible by examining all available evidence and using logical reasoning to conclude that it cannot be achieved or proven. However, there may always be a chance that new evidence or technology could change our understanding of what is possible.
Yes, mathematics can be used to prove that something is impossible in certain scenarios. For example, the proof of the impossibility of trisecting an angle using only a compass and straightedge relies on mathematical principles.
Yes, it is possible for something to be theoretically impossible, but still achievable in practice. For example, it was once thought to be impossible for humans to run a mile in under four minutes, but it has since been achieved by many athletes. So while it was impossible in theory, it was possible in practice with the right training and conditions.