Sequence

A sequence is a function which takes natural numbers as inputs, and returns numbers as outputs.

The Cauchy condition

[TODO I now think this notion is problematic. See below. ]

Suppose that $\{a_{n}\}_{n=1}^{\infty }$ is a sequence that computes π. [TODO what does this even mean? Does it actually make sense? If π is the ratio between a real circle's circumference and a real circle's diameter, then it's not a single number, but some class of numbers. Sometimes the ratio is 3.14, sometimes it's 3.18, etc.]

The practical question to answer, for someone computing this sequence on a computer, is "after how many steps can I stop running the computer program?" The answer is that you should wait until successive steps of the sequence vary by amounts smaller than your desired accuracy.

For example, if your target accuracy is 0.01, and if the printout looks like

$n$	$a_{n}$
1	4
2	3.0
3	3.2
4	3.09
5	3.16
6	3.139
7	3.143
8	3.1408
9	3.1419
10	3.14149
...	...

then we can stop computing around step 9, because at that point the output of the programming is varying by amounts much smaller than 0.01.

But how do we know that later, after many more steps, the sequence won't change in such a way that the earlier digits are affected? E.g. what if the 11th element of the sequence is 3.15? The answer is that what I have done in the above paragraph is I have implicitly assumed that the sequence $\{a_{n}\}_{n=1}^{\infty }$ satisfies the Cauchy condition. A sequence is said to satisfy the Cauchy condition if, for any level of desired accuracy, there is some point beyond which any two outputs of the algorithm differ by an amount smaller than the desired accuracy. More formally, $\{a_{n}\}_{n=1}^{\infty }$ satisfies the Cauchy condition if for any $\epsilon :\mathbb {Q}$ such that $\epsilon >0,$ there exists an $N:\mathbb {N}$ such that if $n,m\geq N$ then $|a_{n}-a_{m}|\leq \epsilon$.

Note that the Cauchy condition is only an idealization of the property that we would actually want a sequence to satisfy in real life. Some Cauchy sequences are not practical: For example, someone could define a Cauchy sequence for which we must choose $N$ to be at least a googolplex whenever $\epsilon =0.1$. Conversely, there are some practical sequences which are not Cauchy: For example, some sequences of partial sums that arise in perturbation theory are divergent, but give accurate and useful results if we only go a dozen terms deep into the sequence.

Examples

The sequence $\{2n+1\}_{n=0}^{\infty }$ could be properly thought of as that concept which---in appropriate contexts---subsumes the following concretes:

• the method that a child uses to produce the nth odd number
• a C++ program which takes in a uint32, n, and outputs 2n+1
• the following table on my screen [a screenshot of a few rows of an excel table where the entries are 2*row + 1 ]
• a succession of locations of a bouncing ball at various times

The problem with Cauchy

All those things are examples of sequences. But are they, or could they be, examples of Cauchy sequences? Unfortunately, we will see that there can be no Cauchy sequences in reality.

Suppose I identify a real life thing $R$ as being a unit of a sequence $\{S_{n}\}$. In my specification of $\{S_{n}\}$ I had to be very precise, so it makes perfect sense to ask whether or not $\{S_{n}\}$ is Cauchy. However, to answer whether or not $\{S_{n}\}$ is Cauchy, I have to go to arbitrarily large $n$; I have to think about parts of $\{S_{n}\}$ that have no actual referent in $R$. So being Cauchy is only a property of the concept $\{S_{n}\}$ itself, not a property of the actual physical concrete thing $R$.

This might seem like a trivial difference, but it's not. It renders the question "is $R$ Cauchy?" meaningless. Worse than that, it also makes the concept of Cauchy useless. Indeed, since any real sequence $R$ is finite, and since sequences are open-ended, I can, in all situations, identify $R$ as being a unit of $\{S'_{n}\}$ and also as being a unit of $\{S_{n}\}$, where $\{S_{n}\}$ is Cauchy and $\{S'_{n}\}$ is not. $\{S_{n}\}$ and $\{S'_{n}\}$ agree for observable values of $n$, which is why they are both potentially valid identifications of $R$.