Welcome to Week 10. The theme this week is to give a precise meaning to the very important concept of convergence of a sequence of real numbers. We will also discuss some other properties of sequences, such as boundedness and monotonicity. We will prove the Monotone Convergence Theorem, which states that the properties of boundedness and montonicity in combination are enough to guarantee that a sequence is convergent – along with the Axiom of Completeness in the real numbers.

In Lecture 20 on Thursday, we will start to look at how we can adapt the idea of convergence for sequences to define a concept of convergence for infinite series or infinite sums. Here is (an old video version of) Lecture 20.