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
- Why do we need this definition?
- 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?
Best Answer
Your main question seems to be
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:
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 $x = \sum_{\alpha \in A}\langle x, b_\alpha \rangle b_\alpha$.