To help you feel more confident about induction, we'll try to prove a couple of statements that we know are wrong, to show you that you can't use induction to prove something that ain't so. But let's try to prove this false statement, and see what happens. Let's try another one. And that's where the induction proof fails in this case. I hope these examples, in showing that induction cannot prove things that are not true, have increased your confidence in the technique.
Induction is actually quite powerful, and more than just a little clever.
Here are a couple more examples, showing induction proving some true statements.