What is the difference between say a sequence $x_{n}$ defined as ,$x_{n} = 1$ for at most finitely many $n$ , $n \in \mathbb{N}$ and $x_{n} = 1$ for at most but finitely many $n$ , $n \in \mathbb{N}$ ?, I think that in the first case $x_{n} = 1$ for only finite values of $n$ and in the second case $x_{n} \neq 1$ for finitely many $n$ , Is this correct ?.
Meaning of $x_{n} = 1 $ for at most finitely many $n$?
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real-analysis
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1"for at most but finitely many" sounds strange. But I think your interpretation is right. It probably meant to say "but finitely many" – 2017-02-10
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1Maybe you meant "for all but finitely many"? – 2017-02-10
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0how about if i replaced "for at most but finitely many $n$" by "for all most but finitely many $n$"? – 2017-02-10
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1It should be "all except finitely many $n$" – 2017-02-10
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0Is this phrase given in English or another language, if it's another language then type it in that language in the comments. – 2017-02-10
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1My two cents. I have never heard "for at most but finitely". It is utterely incomprehensible to me. And I have no idea what they mean. The "but" seems to be unnecessary. "For at most finitely many" means there are zero or finite many. "all but finitely many" mean there are zero or finitely many that are not. I would think "at most finite" and "at most but finite" would mean the same thing. But if "at most but finite" means "all but finite" (the exact opposite) I wouldn't be surprised. – 2017-02-10
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The usual phrasing would be $x_n=1$ for (at most) finitely many $n$, which means that if $n$ is greater than some $N$ we never see $x_n=1$ any more. There is some last $n$ for which $x_n=1$. The usual phrasing of the other would be that $x_n=1$ except for (at most) finitely many $n$. In that case there would be a last value of $n$ for which $x_n \neq 1$. The "at most" could be there to emphasize that there might be no $n$ at all for which $x_n$ satisfies the condition.