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Does a sequence that is eventually constant contain less terms than one that is not?

I don't know how to properly think about this, one could either argue that

a sequence : 1 2 3 1 1 1 1 1 1 1 1 ..., contains as many terms as 1 2 3 4 5 6 7...

or one could say that the sequence only contains 3 terms, whereas the other contains $\mathbb{N}$ many terms

What is the correct way to think about this

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  • What we say is "1231111... has 'only finitely' many 'non-constant' terms". But both sequences have an infinite number of terms. – fleablood Jun 16 '16 at 05:13
  • What does it mean to say that one sequence "contains as many terms" as another sequence? This has not been defined. – littleO Jun 16 '16 at 05:14
  • The sequence $\langle 1,2,3,1,1,1,\ldots\rangle$ has infinitely many terms; it has only three distinct terms, however. More technically, the sequence is a function with domain $\Bbb N$, and the terms $\langle 0,1\rangle,\langle 3,1\rangle,\langle 4,1\rangle,\langle 5,1\rangle$, and so on are clearly not the same object. However, the range of the sequence, i.e., the set of values that it assumes, is ${1,2,3}$, which is finite. This is one reason that it is important always to distinguish a sequence $\langle x_n:n\in\Bbb N\rangle$ from its set of values ${x_n:n\in\Bbb N}$. – Brian M. Scott Jun 16 '16 at 05:17
  • @fleablood "What we say is "1231111... has 'only finitely' many 'non-constant' terms"." Do we say that? Never heard the phrase. – Did Jun 16 '16 at 05:24
  • Really? I've heard and used "only finitely many" frequently and often. ("non-constant terms" maybe not so much but not never). What terminology would you use to describe an infinite sequence where all but finite of the terms are equal. ... "all but finite" might be more common than "only finitely many". In my opinion this are both fairly self-descriptive and clear and common-- we don't need to have a technical and official term for everything after all. – fleablood Jun 16 '16 at 20:29
  • Okay, quick google search. Did is probably right. "all but finite" is more frequently used to describe an infinite sequence, list or set where a finite number are exceptions. "only finitely many" is usually used for slightly different situations where we are enumerating a finite occurence in and of itself and not in context of some other infinite set. Still I don't think either are technical or authorative. – fleablood Jun 16 '16 at 20:35

2 Answers2

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An infinite sequence of real numbers, formally speaking, is a function from the natural numbers to the real numbers. The terms of the sequence are the values of the function, listed in order. That is to say, if your function is given by the rule $$ n \mapsto a_n,$$ then the terms are $$a_1, a_2, a_3, \ldots .$$ The first term and the third term may have equal values, but they represent different places in the sequence, so you should think of them as being different.

treble
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A sequence of real numbers is, by definition, a fuction $x:\mathbb {N}\to\mathbb {R}$. Then, it has always infinite terms.

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