Math for Non-Geeks/Cauchy sequences

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Motivation

In the last chapter, we learned about convergence of sequences $a_n\to a$. They were defined via the Epsilon-criterion, which says that $|a_n-a|<ϵ$ must hold for all but finitely many $a_n$. I.e. it holds for all $n\ge N$ with some $N\in \mathbb{N}$. However, in order to apply this criterion, we need a candidate for the limit $a$. What if we don't have such a candidate? Or more precisely,

A sequence converges, if the distance of $a_n$ to $a$ eventually tends to 0. In that case, also the difference between elements $|a_n-a_m|$ tends to 0. Conversely, if we know that $|a_n-a_m|$ goes to 0 if both $n$ and $m$ get large, then the amount of points where a possible limit or accumulation point $a$ might be shrinks together to a point. So the sequence should converge, then

The interesting question is now, what condition we have to put on $n$ and $m$. If we just consider neighbouring sequence elements via setting $m=n+1$, we might run into trouble: consider the sequence

This is a harmonic series. The difference of neighbouring elements is $|a_n-a_{n+1}|=\frac{1}{n}\to 0$. But the sequence itself diverges to $\mathrm{\infty }$. We need a stronger criterion, i.e. we need to consider more pairs $(n,m)$ than just neighbours.

Derivation of Cauchy sequences

We try to get a condition on $|a_n-a_m|$, which suffices for showing that a sequence converges. So let's take a sequence $(a_n{)}_{n\in \mathbb{N}}$ converging to $a$ and play a bit with it: The epsilon-definition of convergence reads

Template:Math We fix $ϵ>0$. Then, there is an index $N_{ϵ}\in \mathbb{N}$ depending on $ϵ$ with $|a-a_n|<ϵ$ for all $n\ge N_{ϵ}$ . What can we say about the difference $|a_n-a_m|$ for $n,m\ge N_{ϵ}$? There is

So the triangle inequality for $|a_n-a_m|$ implies:

Sequence elements coming after $a_{N_{ϵ}}$ are all separated by less than $2ϵ$ . Visually, all sequence elements are all situated inside the interval $(a-ϵ,a+ϵ)$ which has width $2ϵ$:

The distance between two points in this interval is less than $2ϵ$ . So for a convergent sequence $(a_n{)}_{n\in \mathbb{N}}$ we have:

The $2$, in here is a bit awkward to most mathematicians. They remove it by defining $\widetilde{ϵ}=2ϵ$. The function $x\mapsto 2x$ is mapping ${ℝ}^{+}$ bijectively onto ${ℝ}^{+}$ . So instead saying "for all $ϵ>0$", we could also use the term "for all $\widetilde{ϵ}>0$":

Sequences, which fulfil the above property are called Cauchy sequences . This definition does not require a limit $a$ . A sequence which converges, fulfils the above property, so any convergent sequence is a Cauchy sequence. But seeing that any Cauchy sequence converges is not so easy. Generally, this is even wrong: Not every Cauchy sequence converges! However, it is true that every Cauchy sequence in $ℝ$ converges. In the rest this article , we successively construct a proof for this.

Definition of a Cauchy sequence

File:Zahlen - Quatematik.webm The definition for a Cauchy sequence reads: Math for Non-Geeks: Template:Definition

Intuitively, a sequence is Cauchy, if the difference between any two elements gets arbitrarily small as the element indices go to $\mathrm{\infty }$. Beware: it is a common mistake to think that only neighbouring elements must get close to each other. However, Cauchy sequences require all differences between elements to go to 0:

• 
  A Cauchy sequence: differences between two elements tend to 0.
• 
  Not a Cauchy sequence: differences do not tend to 0.

Every convergent sequence is a Cauchy sequence

We essentially already proved this within the "derivation of Cauchy sequences". But it's always a good idea, to write down one's findings in a structured way. So let's do this:

Cauchy sequences are bounded

Convergent series are bounded. And we can prove the same for Cauchy sequences:

Cauchy sequences with convergent subsequences also converge

We would like to show that in $ℝ$, a Cauchy sequence converges. This is a somewhat longer task. So we first take a smaller step and prove the following smaller theorem:

Every Cauchy sequence converges

File:Cauchy Konvergenz.webm

Now, that we have the smaller theorem above, we can use it to show the final theorem:

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