My first response is that you can break light up into different wavelengths by using a prism. But then I thought, how can you prove that the prism just didn't create a bunch of colors from the color "white?"
Try this easy experiment. Look around when you see things with white light. Do you see colors? That proves that white contains a mixture of colors (unless you are color blind). You see, if you had only a pure red light the only color you could see would be red. Red things would look red, but green things would look black. That's because what green objects do is absorb all the colors except green, which is scattered away from the green object. When there is no green in the light which shines on a green object, there is nothing to scatter back into your eye. So it all gets absorbed, and it looks black to you. A word of caution if you try to check this out. Many colored lights are not purely one color, and many objects are not pure colors either. But if you have reasonably pure colors you will see the effect.