Sequences and the Reals - 13
In this lecture, our goal is to connect our understanding of sequences with one of our big ideas of the course: what makes the real numbers so special? In particular, we focus on two themes. First, we create a link between convergent subsequences and compact sets, which allows us to connect some previously disconnected theorems. Second, we’ll introduce the notion of a Cauchy sequence, which will eventually lead us to a tantalizing new description of metric spaces. We see that convergence of Cauchy sequences gives us a new way to understand the completeness axiom of the reals.
Reading: Rudin Ch. 3: Subsequences, Cauchy Sequences (pg. 52–54)
Pre-Lecture Materials
Sequences and the Reals: Lecture Video
In-Class Activities
Open the following Overleaf link and make a copy for yourself. Remember to include the names of everyone who worked together in the document.
Lecture 13 In-Class Activity: Sequences and the Reals, Completeness
Assignments
- Submit Overleaf Proof 13: Cauchy Sequences
- Homework 7