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This is a question from the set theory section of Kolmogorov & Fomin's Introductory Real Analysis

Let $A_n$ be the set of positive integers divisible by $n$, find the set $$\bigcap\limits_{n=2}^\infty A_n.$$

I would think that the answer is $\{0\}$, but I don't know if $0$ is considered a positive integer in this context. Any help would be appreciated.

Andrés E. Caicedo
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Kidkak
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    $0$ is not a positive integer in any context. It is sometimes considered a natural number. – Joffan Jul 13 '16 at 23:01
  • @Joffan I was referring to this answer, which says that zero is positive (and negative) in typical French usage. – Kidkak Jul 13 '16 at 23:26
  • @Elliot I don't understand your comment. Do you mean the answer is not 0? – Kidkak Jul 13 '16 at 23:28
  • I was trying to think of the name... the law of trichotomy - Every real number is either negative, zero, or positive. --- As for the French usage, I'm guessing that the book you refer to is not written in French. – Joffan Jul 13 '16 at 23:31
  • It was originally written in Russian, and having read that about French usage it seemed reasonable not to make the assumption it is not the case in Russian. – Kidkak Jul 13 '16 at 23:34
  • If you consider 0 to be positive (which no-one does) the answer is {0} if you do not consider zero to be positive the answer is the empty set. There's no profundity to the answer either way. If A_n were defined to be all integers, or all Natural numbers or all Natural numbers including zero the answers would have different answers but relevent "moral" would be the same: 0 is the only integer that is divisible by all non-zero integers. – fleablood Jul 14 '16 at 00:04
  • "I would think that the answer is {0}, but I don't know if 0 is considered a positive integer in this context." So what do you think the answer would be if 0 isn't considered a positive integer (which it isn't)? – fleablood Jul 14 '16 at 00:05

2 Answers2

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$$\bigcap\limits_{n=2}^\infty A_n=∅$$ because there is no any positive integer that is divisible by all integers without a remainder.

ibnAbu
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0 is not positive.

$A_i \subset \mathbb Z^+$ so $\cap A_n \subset \mathbb Z^+$

Let $k \in \mathbb Z^+$. $k + 1$ does not divide $k$. So $k \not \in A_{n+1}$ so $k \not \in \cap A_n$. So $\cap A_n$ contains no positive integers but contains nothing else either.

So $\cap A_n = \emptyset$.

Let $n \ge 2$

fleablood
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