I see, very helpful, thanks. But doesn't this leave room for what you potentially have not accounted for? I am just trying to wrap my head around how one can write a proof by saying something doesn't occur
Good question. That is part of the second point. You have to show that there is no room left.
For example say you want to show that a real valued solution stays bounded. Then you have to show that a solution always exists, starts at some small value and "it never occurs that the absolute value of the solution is bigger than 1000". Because you ruled out other scenarios this then implies that the solution is always bounded by 1000.
For example say you want to show that a real valued solution stays bounded. Then you have to show that a solution always exists, starts at some small value and "it never occurs that the absolute value of the solution is bigger than 1000". Because you ruled out other scenarios this then implies that the solution is always bounded by 1000.