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We know that $\mathbb{Q}$ is countable so let $\{q_n\}_{n \in \mathbb{N}}$ a sequence of all rational numbers.

My Question is:

Is any subsequence $\{q_{n_k}\}_{k \in \mathbb{N}}$ dense in $\mathbb{R}$

Thanks.

Matey Math
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2 Answers2

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"Any". Obviously. The sequence itself is a subsequence. And if you meant proper sequence just remove one term. Or take the sequence of those with even denominators, every neighborhood of a real contain a rational with an odd denominator.

"All". Obviously not. Take the $q_k$ that are integers.

fleablood
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  • +1 I was about to write a comment (as if the existing comments were not enough . . .), but after reading your answer, I think it both answers the question and addresses the ambiguity issues. It's fairly well known (or so I thought) that "any" is often ambiguous and one should mostly avoid its use in formal mathematical statements, and this question gives another example of the kinds of things discussed [In proofs, are “for each” and “for any” synonyms?](https://math.stackexchange.com/questions/2696959/in-proofs-are-for-each-and-for-any-synonyms) – Dave L. Renfro Jun 16 '18 at 18:01
  • FWIW I take "any" literally. Is *any* subsequence dense? Well, obviously. It is it's own subsequence and it is dense... – fleablood Jun 16 '18 at 19:35
  • The problem is that "any" can sometimes mean "there exists", and "any" can sometimes mean "for all". This is discussed in the links I gave in the question I linked to in my previous comment. See *Be careful with your use of "any"* [here](http://www.jmilne.org/math/words.html) and see *#13* [here](https://cameroncounts.wordpress.com/2011/07/23/how-to-write-mathematics/). – Dave L. Renfro Jun 16 '18 at 20:45
  • Ah, I see what you mean. Yes that should be avoided. – fleablood Jun 16 '18 at 23:53
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This is equivalent to asking is any subset of $\mathbb Q$ dense?.

Some subsets are dense some are not e.g. $\mathbb N$

zwim
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