In the last few weeks, my friend and I have been reading Pugh’s Real Mathematical Analysis to get a better grasp on the subject. In the next few posts, I’ll be writing about concepts I personally struggled with and ideas I particularly enjoyed.
Dedekind cuts, proving that there exists a number system that satisfies the least upper bound property– that given any upper-bounded set S, there is a number n such that \(n > s \forall s \in S\). In the book we used in the school year, this theorem was just taken to be true. Dedekind cuts provides us with a way of constructing a set of numbers, the real numbers, that satisfies this property. Dedekind cuts also provides us with a way to prove completeness in the sense that if we try to make a cut, we’ll end up with another real number. Something that initially confused me was the different definitions of completeness which can also be expressed in terms of continuity and cauchy sequences. connectedness/