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Possible duplicate: Sequences with Real Indices

I am trying to understand the motivation for various definitions in real analysis. Take for instance the definition of a sequence where it is defined as a function from natural numbers to that of real numbers. But why natural numbers? I cannot substantiate it with a reasonable argument and I am looking for one.

I have been able to substantiate certain definitions based on some of my own reasoning, while that might not have been the real reason why it was defined that way. Take for instance the definition of convergence of a sequence. I think one of the central definitions in analysis is that of convergence of a sequence. Here we say that a sequence is said to converge to a limit $L$ if any $\epsilon$-neighborhood of $L$ has all but finitely many terms of the sequence. I ask the usual set of questions to myself

  1. Why do we need this definition?
  2. Why was it defined this way?

I thought of some possible answers. There are questions about the operations of addition and subtraction, and the manipulation of those operations resulting in questions about re-arrangements of infinite series. The foundation to answering such questions lies in the answer to "does it even even make sense to represent the sum of infinite numbers by one number?" What is the logical basis to represent that infinite sum by one number? Then we get the answer that the sequence of partial sums converge and so we can represent them as one number. Then we get the questions "what is a sequence and what does it mean to say that the sequence converges?" Then we get the definitions for a sequence and the definition of convergence.

Definition of convergence makes sense to me. Let's assume we are thinking of two numbers $a$ and $b$. What are some of the rigorous ways to define the quality of two numbers being equal? A popular one is to say that $a \geq b$ and $b \geq a$. Another is the topologic definition where we say that $a$ always lies in any $\epsilon$-neighborhood of $b$ no matter how small $\epsilon$ is. Now, we can modify this definition to obtain the definition of convergence of a sequence where we replace "$a$" with "all but finite terms of the sequence". There you go. We got a definition for the convergence of a sequence.

The part I could not substantiate to myself was the need to define sequences as functions on natural numbers to real numbers. Why not define sequences on some other index set? For instance, why not define it on real numbers? More generally, we have countability properties defined but why even bother with natural numbers when what we want in the end is real numbers?

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    $\begingroup$ "why natural numbers? " It is the "obvious definition of sequence : one element after another. Then - as usual in math - we generalize it. $\endgroup$ – Mauro ALLEGRANZA Jul 27 at 9:10
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    $\begingroup$ a "sequence" defined on the real numbers is a function $\Bbb R\to\Bbb R$. Of course these are widely studied. $\endgroup$ – Angina Seng Jul 27 at 9:11
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    $\begingroup$ A "sequence" is just a fancy word for a function whose domain is the natural numbers. Mathematicians introduce such fancy words when they meet a concept that appears over and over again and they want to reduce the number of characters needed to describe it. As it happens, functions whose domain is the set of natural numbers appear EXTREMELY often, so this compression scheme turns out to be very efficient. In fact, even the word "function" is just a shortcut for "a relation on $X\times Y$ that satisfies such and such properties...", so the compression happens at many levels. $\endgroup$ – Michal Adamaszek Jul 27 at 9:25
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    $\begingroup$ If you really want to understand mathematics you have to know something about its long history which explains where we are today. Read books on the history of mathematics. For example, From One to Zero: A Universal History of Numbers by Georges Ifrah. $\endgroup$ – Somos Jul 27 at 11:52
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    $\begingroup$ @Somos: Regarding Ifrah's books, I've read several not very flattering reviews of his books. For example, see the 2 reviews by Joseph Dauben (review 1 and review 2; recall that Dauben is the author of a well-known biography of Cantor) in Notices of the American Mathematical Society and the review by Eli Maor in American Mathematical Monthly. $\endgroup$ – Dave L. Renfro Jul 27 at 15:46
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When you have a sequence of real numbers, there is a first number in this sequence. Give it the index 1. Then there is a second number, give it the index 2. Continue. You might now ask: what is the real number with index $n$ in my sequence? For this, you define the function $f:\mathbb{N}\rightarrow\mathbb{R}$ which maps indices to the corresponding real numbers in your sequence.

When you now think a bit about it, your sequence is completely defined by the function $f$, and on the other hand $f$ is completely defined by the sequence. In other words, there is a bijection of sequences of real numbers and functions $\mathbb{N}\rightarrow\mathbb{R}$. This is the reason why we say that $f$ itself is the sequence.

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  • $\begingroup$ I suppose that the motivation to use $\mathbb{N}$ comes from the fact that we need the indices to have the quality of having a "next element." This way, any dense set will not work as an index set for the purpose of defining a sequence. $\endgroup$ – TryingHardToBecomeAGoodPrSlvr Jul 27 at 10:20
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    $\begingroup$ Yes, you need a next element, but you also need a first element. Thus, $\mathbb{Z}$ doesn't work. But you could as well use $\mathbb{N}\cup\{0\}$ or $\{x\in\mathbb{Z}:x>-12345\}$. $\endgroup$ – NeitherNor Jul 27 at 10:26
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    $\begingroup$ Oh, and before I get criticized: one can of course argue which one would be the more general definition. The point is that, in the common definition, a sequence starts somewhere, and thus we should index the elements by something which has a "smallest element". $\endgroup$ – NeitherNor Jul 27 at 10:57
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    $\begingroup$ @TryingHardToBecomeAGoodPrSlvr Perhaps it would be good enough to use any well-ordered set as index set / domain of the function. So just ake an ordinal. The smaller the better. Then again, a finite ordinal leads only to the less interesting notion of a tuple. So, just pick the smallest infinite ordinal ... and here we are $\endgroup$ – Hagen von Eitzen Jul 28 at 20:50
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    $\begingroup$ @TryingHard it's not really that strange. Take a net with index set $A=\mathbb{R}$ and topological space $X=\mathbb{R}$. So we now have... yes, the good old functions $f:\mathbb{R}\rightarrow\mathbb{R}$ from school. And the fancy neighborhood convergence rule becomes there good old $\delta-\epsilon$ convergence rule saying e.g. that $\frac{1}{x}\rightarrow 0$ for $x\rightarrow\infty$. And suddenly you realize that convergence of sequences and convergence of functions is really the same thing. And that is the beauty of abstract concepts in math. $\endgroup$ – NeitherNor Jul 29 at 12:15
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Your main question seems to be

Why not define sequences on some other index set?

Okay, the term sequence is reserved for functions $a : \mathbb N \to X$ (note that I wrote $X$ instead of $\mathbb R$ because we may consider sequences in arbitrary sets, or if we are interested in convergence, in arbitrary topological spaces). However, this can easily be generalized to other index sets. This results in the concept of a net. See here. Quote:

Let $A$ be a directed set with preorder relation $\ge$ and $X$ be a topological space. A function $f: A \to X$ is said to be a net.

Recall that a preorder relation $\ge$ on a nonempty set $A$ is a reflexive and transitive binary relation. We do not require it to be antisymmetric (if it is, it is called a partial order). $(A,\ge)$ is called directed if for all $a, b \in A$ there exists $c \in A$ with $c \ge a$ and $c \ge b$. Simple examples are $\mathbb N, \mathbb Z, \mathbb Q, \mathbb R$ with their natural partial order. These sets are even totally ordered which means that for any two elements $a, b$ we have $a \ge b$ or $b \ge a$.

We often write nets in the form $(x_\alpha) = (x_\alpha)_{\alpha \in A}$. A net $(x_\alpha)$ is said to convergence to $x \in X$ if each open neigborhood $U$ of $x$ in $X$ admits $\alpha_0 \in A$ such that $x_\alpha \in U$ for all $\alpha \ge \alpha_0$. This obviously extends the concept of convergence from sequences to more general objects.

Interesting examples of nets which are no sequences occur in calculus when the Riemann integral is introduced. Here we consider partitions $\mathfrak{P} = (t_0, \ldots,t_n)$ of an interval $[a,b]$ and order them by"set inclusion" (i.e. $\mathfrak{P}' \ge \mathfrak{P}$ if $\{ t'_0, \ldots,t'_{n'} \} \supset \{ t_0, \ldots,t_n \}$). The uncountable set $P$ of all partitions is a directed partially ordered set, but it is not totally ordered. For any bounded function $f : [a,b] \to \mathbb R$ we consider the lower and upper Riemann sums which gives us two nets indexed by $P$. Both nets converge, but in general do not have the same limit. If they have, then $f$ is called Riemann integrable and the common limit is called the Riemann integral of $f$.

However, if you study calculus textbooks, you will find that frequently one again works with suitable sequences of partitions, e.g. equidistant partitions with mesh $(b-a)/n$. But note that conceptually one has to go beyond sequences.

An equivalent approach to the Riemann integral is by taking the elements of a modified index set $\mathbf P$ to be all systems $(\mathfrak{P},\mathbf x)$ where $\mathfrak{P} = (t_0, \ldots,t_n)$ is a partition of $[a,b]$ and $\mathbf x = (x_1,\ldots,x_n)$ with $x_i \in [t_{i-1},t_i]$. Defining $(\mathfrak{P}',\mathbf x') \ge (\mathfrak{P},\mathbf x)$ iff $\mathfrak{P}' \ge \mathfrak{P}$ we get a preorder on $\mathbf P$ which is directed but not antisymmetric. For any function $f : [a,b] \to \mathbb R$ we consider the Riemann sums $R(f;\mathfrak{P},\mathbf x) = \sum_{i=1}^n f(x_i)(t_i - t_{i-1})$. This system is a net over $\mathbf P$, and it is well-known that this net converges iff $f$ is Riemann integrable (and in that case the limit of the net is the Riemann integral of $f$). This shows once again that nets which look at first glance "fairly exotic" occur quite naturally in calculus and that we do need a concept of convergence for such nets.

So what is the reason for the prominent role of sequences in calculus? First of all, they are much simpler than general nets. For example, you can use induction. But the deeper reason is that all topological properties of $\mathbb R$ can be expressed via sequences. For example, $M \subset \mathbb R$ is closed iff for all sequences $(x_n)$ in $M$ which converge to some $x \in \mathbb R$ one has $x \in M$. Moreover, $M \subset \mathbb R$ is compact iff each sequence $(x_n)$ in $M$ has a convergent subsequence. Also the continuity of functions can be expressed via sequences: A function $f : \mathbb R \to \mathbb R$ is continous at $x$ iff for all sequences $(x_n)$ such that $x_n \to x$ one has $f(x_n) \to f(x)$.

Let me finally remark that series $\sum_{n=1}^\infty a_n$ should (at least in my opinion) be treated as nets. The usual approach is to consider the sequence $(s_m)$ of partial sums $s_m = \sum_{n=1}^m a_n$ and study its convergence. I suggest to introduce the set $\mathbf F$ of all finite subsets of $\mathbb N$ and order it by inclusion ($F' \ge F$ if $F' \supset F$). This produces a directed partially ordered set $\mathbf F$. Define a net over $\mathbf F$ by $(\sum_{n\in F}a_n)_{F \in \mathbf F}$. This net may converge or not. If it converges, we denote its limit by $\sum_{n\in \mathbb N} a_n$. It is now a nice exercise to show that $\sum_{n\in \mathbb N} a_n$ exists iff $\sum_{n=1}^\infty a_n$ is unconditionally convergent. Perhaps this unusual approach seems to be too exotic, but it has the benefit that it can be generalized to sums over uncountable index sets. Such sums occur in the context of Hilbert spaces. It is well-known that each Hilbert space $H$ has a (possibly uncoutable) orthonormal basis $\{b_\alpha\}_{\alpha \in A}$ and that each $x \in H$ can be written as $x = \sum_{\alpha \in A}\langle x, b_\alpha \rangle b_\alpha$.

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    $\begingroup$ The motivation for defining nets is not that Riemann integrals are nets, but that in arbitrary topological spaces, sequences are not adequate to define convergence, compactness, and continuity: some topological spaces are "too big" for their topological features to be studied with sequences alone, but you can use nets instead of sequences and the language carries over well, e.g., an arbitrary topological space is compact iff each net in it has a convergent subnet. A catch here is that the definition of subnet is subtle! In metric spaces, sequences are adequate. $\endgroup$ – KCd Jul 28 at 1:27
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    $\begingroup$ @KCd You are right concerning arbitrary topological spaces. The concept of a net, however, arose in analysis. It was introduced by E.L.Moore and H.L. Smith in 1922. See here. The concept was generalized to arbitrary topological spaces by G. Birkhoff in 1937. See here. $\endgroup$ – Paul Frost Jul 28 at 8:17
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    $\begingroup$ Thanks for the correction to what I had said. What is the setting of "General Analysis" that Moore and Smith write about, if it is not convergence in arbitrary topological spaces? $\endgroup$ – KCd Jul 28 at 9:25
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    $\begingroup$ @KCd Good question - what is General Analysis? You can read this, but it does make it really clear. Quote: "Those theories of Analysis in which at least certain of the functions involved are on a range $\mathfrak P$ of elements $p$ of a nature not specified, or at lleast not fully specified, we may designate as theories of General Analysis". Whatever $\mathfrak P$ may be, the concept of convergence introduced by Moore is only for real-valued nets (or possibly for nets in normed linear spaces). $\endgroup$ – Paul Frost Jul 28 at 9:45
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    $\begingroup$ I looked at your links and some other references, and it appears that "General Analysis" is Moore's name for what is essentially functional analysis using Moore's idiosyncratic notation. For example, he said the inspiration for General Analysis is similarities among systems of linear equations, integral equations, and Hilbert spaces (or "Hilbert's theory of quadratic forms in infinitely many variables"). $\endgroup$ – KCd Jul 28 at 14:40
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Depends what you want/need. In applications as far as most of us are aware at least, the index will come from a countable set as you say.

For example thinking of annuities (amounts of money payed at regular intervals), we usually think of these as payments at discrete times for obvious reasons (possibly infinite) and so we would be treating them as a sequence of payments indexed by naturals (can apply to it the ‘usual‘ sequence theory).

If the payments are approximated/thought of as ‘continuous’ (indexed by $\mathbb{R}$ if you wish), then we use calculus (integration).

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When you’re doing real analysis, you need an ordered set of points. That is, $x_1$ comes before $x_2$ which comes before $x_3$ and so on. (Whether or not $x_1<x_2$ is not what I’m talking about here.) That means that the indices 1, 2, 3 $\ldots$ are ordered. More precisely, we have a map $x:I\to X$ where $I$ is the index set and $X$ might be ${\mathbb R}$ or ${\mathbb R}^2$ or something. What we need is for $I$ to be ordered. We also need $I$ to be infinite, otherwise we can’t take limits or talk about convergence. Now ${\mathbb N}$ is the least infinite totally ordered set. Even if we don’t require a total order, we can’t talk about convergence unless $I$ contains ${\mathbb N}$ as an ordered subset, so ${\mathbb N}$ is universally minimal for our purposes. In other words, we’re not using ${\mathbb N}$ in its role as a monoid (a subset of the additive group ${\mathbb Z}$), we’re using ${\mathbb N}$ in its role as the least infinite ordinal, $\omega$.

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Simply put, because:

$$a_n$$ $$n = 0, 1, 2 ...$$

can be thought of as

$$a(n)$$ $$n = 0, 1, 2 ...$$

which is just a function on the natural numbers.

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