Suppose $x\in \mathbb \cap_{k=1}^{\infty} \cup_{n>k} \{x:x=nk\}$. Is $x$ belonging to the latter intersection/union equivalent to saying that for each natural number $k$, there exists a natural number $n$ bigger than $k$ such that $x$ equals $n$ times $k$? Thanks.
The intersection of the union of sets
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real-analysis
elementary-set-theory
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0Yes (provided by "natural number" you mean a positive integer, rather than a non-negative integer). – 2017-02-26
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0And answer will be "Yes" regardless of what you want to say about $x$. Since the phrase "Suppose $x\in \mathbb \cap_{k=1}^{\infty} \cup_{n>k} \{x:x=nk\}$ " means $x\in\varnothing$. See https://en.wikipedia.org/wiki/Vacuous_truth – 2017-02-26
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0Thanks, NCh, so you are saying that no matter what set I construct in the above way, no x's will belong to it? – 2017-02-26
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0@Socchi: Depending on what "that way" means exactly, there are non-empty sets you can construct that way. See [Wikipedia](https://en.wikipedia.org/wiki/Set-theoretic_limit) for details. – 2017-02-26
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0celtschk, do you mean that x belonging to the intersection of the union of the set of x such that x=nk is not equivalent to the statement that given any natural number k there exists a natural number n>k such that x=nk? If so, where is the flaw in my reasoning? – 2017-02-26
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0@Socchi: Yes. (I deleted my comment because the specific way I formulated it, it is not true). The set you constructed is empty. Since you didn't give the reasoning that led you to that expression, I cannot tell where the error is in that reasoning. – 2017-02-26
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0oh alright yes I see, but in general the equivalence of the two statement would hold, although it could be that no element is actually in that set like in this case,correct? – 2017-02-26