For some problems, it is difficult to get an explicit solution. But it is very good if we can prove there do exist some solutions though we can't find them. Is it possible to prove existence of solutions without constructing one? Can you show some examples? Many thanks.
Edit: Thanks everyone. Of course, it is the best if we can find one or all solutions to a problem. But it is not always easy. Then a non-constructive proof may also be very meaningful. Here is an example. In 1991, it was proved that a multiple layer perceptron neural network is a universal function approximator. But the proof is not constructive. However, it is of great importance for at least control community as people can try to use neural network to approximate any nonlinear systems without worrying about the theoretical foundation. What I am interested is whether there are any general methods for non-constructive existence proof.