How does proof by induction work?

If I gave you an infinitely tall ladder and asked you to prove to me that there are no rungs missing from the ladder, you wouldn't start to check the ladder rung by rung to see if they are all there, you'd never finish! A better way is to prove (this is much more straightforward for mathematical statements than ladders!) that if a rung is in place then the next rung up is also in place. If you managed to do that, then all you have to show me is that the bottom rung is there! Then from your proof I will know that the next rung up is also there, and since that one is there, I'd know that the one above is also there, and so on... This is 'in essence' how proof by induction works - replace the ladder with a mathematical statement depending on an integer N, and the rungs with N=0, 1, 2, 3... You prove that that if the statement is true for an integer, it's also true for the integer above that. Then just show that it's true for the lowest integer case - and you've proven 'by induction' that it's true for all the integers above that.