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In every Real Analysis text that I've seen, the word 'sequence' always means "an infinite list of real numbers". Why is it so useful that the definition includes "infinite"? I have a similar issue with anything that doesn't distinguish between infinite series and finite series.

Jean Marie
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J. Min
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    For one, the properties of sequences and series that one usually cares about in analysis (various flavors of convergence, rates thereof, etc.) only make sense for infinite sequences/series, anyway. – Travis Willse Apr 25 '16 at 18:17
  • There are other areas of mathematics (outside of analysis) that use finite sequences more frequently than infinite sequences. – Michael Burr Apr 25 '16 at 18:18
  • Sequences are the mathematician's microscope. E.g., plug in a sequence with certain properties in a function an see what comes out. And finite sequences don't have enough interesting properties for that purpose. – Michael Hoppe Apr 25 '16 at 18:43
  • If by "list" you mean a set then 'sequence' doesn't always mean "an infinite list of real numbers", it just means "a list of real numbers". For example, consider any constant sequence of real number. –  Apr 26 '16 at 04:35

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This is a soft question so I will answer it this way. This is largely based on my personal experience.

First, when people say sequence, they really mean a mapping from the set of all natural numbers to the real numbers. That is, for any $n\in\mathbb{R}$, there exists a corresponding real number $x_n$. Since there are infinitely many natural numbers, sequence has to be infinite in this sense.

But wait a second, it makes perfect sense talking about "finite sequence"! You just restrict your attention on a finite subset of $\mathbb{N}$, for example, $\{0,1,2,3,\cdots,27\}$. But you don't hear people talking about these kind of sequences often. There are some reasons behind it.

In real analysis, at least in undergraduate, infinite sequence really is the core topic. First, analysis is the study of limit, in some sense. When studying limit, sequence is a powerful tool that we can use to understand more complicated topics simply because sequence is the simplest object out there. Most textbooks start by talking about sequence, then you use sequence to study series and functions.

If you think about it, series are complicated things: how the heck can we understand an infinite sum? After all, what is an infinite sum? Does it even make sense? By defining partial sum of a series, we can basically treat series as limit of sequences. Functions are complicated objects, too. They can be continuous or discontinuous, differentiable or nondifferentiable. By defining various properties of functions using sequence, we are basically reducing an unknown problem and unknown object to a known, well-studied object.

3x89g2
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I am not a historian, but I think it's reasonable to assume the concept of sequences developed initially in connection with the concept of approximation. Take the area of a circle, for example. Take regular polygons with $n$ sides, inscribed in the circle. The area of each polygon is an approximation for the area of the circle, but to make real sense of this, you must allow $n$ to run through all natural numbers, you can't just limit it to finitely many $n$'s.