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?