Chapter 2 Sequences

In the previous section we introduced the completeness axiom and used this to define the real numbers $\mathbb{R}$. This is an important milestone: it allows us to talk about the continuum, the idea that numbers form a continuous, unbroken line with no gaps or holes. It also allows us to define new numbers which are not available in the more primitive rational number system, such as $\sqrt{2}$.

However, it is too early to celebrate! So far, our only description of $\sqrt{2}$ is that it equals the supremum of the set $\{a\in\mathbb{R}:a^{2}\leq 2\}$. This is rather abstract and does not appear related to our more familiar (albeit less concrete) understanding of numbers through decimal expansions. To understand how to work with real numbers in practice, we will study sequences and limits.

Sequences are fundamental to our understanding of real numbers. For instance, they will enable us to understand what is really meant by the decimal expansions such as $\sqrt{2}=1.414213\dots$ and to create tools for computing them. However, sequences play a much bigger role in mathematics and have a huge variety of applications: you shall encounter many throughout your degree. For this reason, here we develop a general and robust theory of sequences, which will stand us in good stead for what is to come.

Learning outcomes.

• •
  ​

  Explain key quantified statements such as the definition of a limit of a sequence, in both intuitive and formal mathematical terms. Apply and illustrate definitions through worked examples.

• •
  ​

  Apply theory (such as the limit laws, squeeze theorem, the boundedness and subsequence tests or monotone convergence theorem) to deduce whether a sequence converges and, in cases of convergence, to deduce the value of the limit.

• •
  ​

  Explain the proofs of key results such as the limit laws, monotone convergence theorem, Bolzano—Weierstrass theorem and Cauchy criterion.

• •
  ​

  Use abstract $\varepsilon$-$N$ arguments to prove new statements about convergent sequences.

• •
  ​

  Create examples to illustrate different possible behaviours of sequences.