Can someone prove that nth harmonics has n-1 node?

Take a simple example, the second harmonic (where n = 2) sin (2*pi*x/L). This is zero at x = 0, L/2 and L. Here there is one (n-1) internal node as there should be. This is easily generalized to show that in general there are (n+1) nodes including the end-points or (n-1) excluding them.