I understand how to prove something by induction, but I don't really understand how it works. I understand that you must prove the base step, and then prove the inductive step. For instance, in proving De Moivre's Theorem, you can prove the base step of n=0. And then you can assume that the theorem is true for some integer k, so: (cos(x)+isin(x))k=cos(kx)+isin(kx). The part that confuses me is why can we just assume that the theorem is true for some integer k? Why do we even need to prove something like De Moivre's Theorem if we just can just assume that it is true for some integer k. I've always had trouble wrapping my mind around this concept, so any clarifications would be greatly appreciated.