4.5-4.6 and 6.1, due on September 21

The principle of mathematical induction was tough. It made sense, but many of the proofs are a little tough to follow. I hope we can review this in class, especially Theorem 6.2. I think that they're proving this by contradiction, however it states that if the theorem is false, then it satisfies conditions 1 and 2 (first case is true and the implication is true) but then it states that there's still positive integers for which P(n) is false. If the initial conditions are satisfied by assuming the end is false, you don't have a proof. One of the 2 conditions needs to be false. Also the concept of well-ordering could be explained a little better. I'm not sure how being well-ordered leads to the concept of induction. 

The first two sections, 4.5 and 4.6, were very basic, but useful. Those proofs were short and clever, but nothing really new. Induction is a neat principle, one that I've also had a little experience with. But the proofs are super long! Also mathematically intensive! Good thing I like math. I also liked the way to compute sums super quickly with the little story thrown in there too. I always wondered how people did those big numbers so fast. It's just neat that we can prove stuff like that and then use it!