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$3.1.1$ Definition A sequence of real numbers (or a sequence in $ \mathbb R$.) is a function defined on the set $ \mathbb N$ = { $1 , 2, . . .$ } of natural numbers whose range is contained in the set $ \mathbb R$ of real numbers.

This definition is from the book of 'Introduction to real analysis', and I'm kinda confused about this definition. If it's a function from $ \mathbb N$ to $ \mathbb R$, then the terms of the sequence should be infintely many($a_1,a_2,a_3...$), since $\mathbb N$ has infinitely many numbers of elements. My questions are: Is this a definition of infinite sequence? If yes,why does this book just define a sequence as an infinite one? Thank you!

Andrew Li
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    Not following. Yes, there are infinitely many natural numbers so therefore there could be infinitely many values for any given entry in your sequence. Is that what you are asking? – lulu Aug 11 '22 at 00:35
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    That (correct) definition of a sequence nowhere says that the values of the function need be distinct. Setting $5 = f(1) = f(2) = f(3) = \cdots$ defines the sequence commonlyu written $5,5,5, \cdots$. – Ethan Bolker Aug 11 '22 at 00:35
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    Shortly, yes. This is the definition of an infinite sequence of real numbers. A finite sequence is often defined as a map of a (finite) set of the form ${1,2,\ldots,n}$ to $\mathbb R$, for some $n\in\mathbb N$. –  Aug 11 '22 at 00:39
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    In addition: Sometimes we want to define a sequence where the (infinite) index set starts somewhere other than $1$. – GEdgar Aug 11 '22 at 00:47
  • Thank you @StinkingBishop. I'm just curious how you describe a finite sequence if you define a sequence as an infinite one. – Andrew Li Aug 11 '22 at 00:50
  • for instance negative integers? @GEdgar – Andrew Li Aug 11 '22 at 00:53
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    If we had infinite time and space (for writing) we would always express ourselves as precisely as possible. We would always qualify the word "sequence" with the adjective "finite" or "infinite" and would always say exactly what the domain and range are, etc. etc. We don't usually have that luxury. When you encounter the word "sequence" unqualified, it usually means "infinite sequence", for brevity, unless (of course) it is clear from the context that the author meant something else. –  Aug 11 '22 at 00:59
  • For instance, the sequence $\frac{1}{\log n}$ starts at $n=2$. – GEdgar Aug 11 '22 at 01:02
  • Thank you@SuzuHirose – Andrew Li Aug 11 '22 at 01:16

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The definition provided does require infinitely many terms in the sequence. I'm not sure where the confusion is coming from though, since the definition doesn't promise that the sequence may be finite. In this definition we are taking all sequences to be infinite sequences. One could also define finite sequences, but the book is just not trying to do that.

Addem
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