4.2 through the middle of page 144, due November 7

I have never seen an approximation for factorial and I've never thought about one being useful because I often forget to think about these concepts to really large numbers where it would be difficult to compute them. I'm curious how good of an approximation it is. I think the following lemmas might be getting at that, but I'm a little confused about where they came from (even with the proof) and how to interpret them. I have really lost sight of the big picture, I know we talked about Chebychev's estimation last time and so of course we read the (very confusing) proof today, but the other things seem somewhat random. I know they somehow connect because we keep learning lots of lemmas in order to prove theorems, but they still seem out of place to me.