Compact Sets and the Real Numbers - 10
In this lecture, we finally prove that the interval [a, b] ⊂ R is compact. This result leads us to two even more satisfying theorems that illuminate the nature of compact sets in the reals. The Heine–Borel theorem tells us that what we hoped was true in general is in fact true in the reals: a set is compact in the reals if and only if it is closed and bounded. We’ll also see an implication for limit points in the reals known as the Bolzano–Weierstrass theorem.
Reading: Rudin Chapter 2: Compact sets (pg. 39–40)
Pre-Lecture Materials
- Compact sets and the real numbers: pre-lecture reading
- Compact sets and the real numbers: Full lecture notes
Compact sets and the real numbers: Pre-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 10 In-Class Activity: Heine Borel & Bolzano Weierstrass Theorems
Assignments
- Submit Overleaf Proof 10: Heine-Borel Theorem
- Homework 5