If something proves to be true or to have a particular quality, it becomes clear after a period of time that it is true or has that quality.

If you prove that something is true, you show by means of argument or evidence that it is definitely true.

If you prove yourself to have a certain good quality, you show by your actions that you have it.

If you prove a point, you show other people that you know something or can do something, although your action may have no other purpose.