1.3 Cauchy Sequences

To conclude the lecture recap, we discuss another type of sequence known as a Cauchy sequence1.

Definition 1.2: (Cauchy Sequence)
A real sequence \((a_n)_n\) is called a Cauchy sequence if \[\forall \epsilon >0, \exists N \in \mathbb{N} \; \text{such that} \; \forall n,m \geq N, \; \lvert a_n - a_m\rvert < \epsilon.\]

The idea here is that however close you want members of the sequence to get, say a distance \(\epsilon\), there will always be a point in the sequence after which any two members of the sequence are within a distance \(\epsilon\) of each other.

You’ve already met some Cauchy sequences. For example, all convergent sequences are Cauchy sequences. You might also be tempted to think that the converse holds, that all Cauchy sequences are convergent. Working in the real numbers, you would be right.

Theorem 1.4:

Any real sequence \((a_n)_n\) is convergent if and only if it is a Cauchy sequence.

So, if these things are the same in \(\mathbb{R}\), why do we make the distinction? Suppose we were only working in \(\mathbb{Q}\), and we were given the sequence \(1,1.4,1.41,1.414,1.4142,\ldots\). In \(\mathbb{R}\), this converges to \(\sqrt{2}\), so by our theorem above, this sequence is a Cauchy sequence in \(\mathbb{R}\). As each member of the sequence is rational, we then know that the sequence is a Cauchy sequence in \(\mathbb{Q}\). But since \(\sqrt{2} \not\in \mathbb{Q}\), it is not a convergent sequence in \(\mathbb{Q}\)!

This example shows why the Completeness Axiom is so important! Assuming this axiom allows us to take limits in \(\mathbb{R}\), and its why we’re studying real analysis this semester (and not rational analysis!)

  1. This is one of (way too) many things in analysis named after Cauchy. If you take Analysis 2B next year, you’ll find that most of complex analysis — analysis in \(\mathbb{C}\) instead of \(\mathbb{R}\) — is due to Cauchy.↩︎