I have the following problem:

Let $(a_n)$ and $(b_n)$ two sequences of natural numbers such that, $a_n\to +\infty$ and $b_n\to +\infty$. Prove that: $$K=\{\pm\frac {a_n}{b_m}:n,m\in\mathbb{N}\}$$ is a dense subset of $\mathbb{R}$.

In consequence, prove that the quotients of two primes are dense in the positive reals.

I know that I can prove the second part like in: "Rationals of the form $\frac{p}{q}$ where $p,q$ are primes in $[a,b]$", but I was wondering if I could prove without the prime number theorem.

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    $\begingroup$ It is not true. Let $a_n=10^n, b_n=10^n$, then $\frac{a_n}{b_m} = 10^{n-m}$, and so the corresponding $K$ is not dense in $\mathbb{R}$. Zero must be in the closure, of course. $\endgroup$ – copper.hat Aug 28 '13 at 23:18
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    $\begingroup$ I wonder what extra conditions might be imposed to imply the desired density. (Maybe another question could be put about that.) $\endgroup$ – coffeemath Aug 28 '13 at 23:32
  • $\begingroup$ Where did you get this problem (they are asking you to prove something that is not true)? $\endgroup$ – Stefan Smith Aug 28 '13 at 23:34

This is false. For example, take $a_n=b_n=2^n$. Then $K$ consists of $\pm$ all powers of $2$ to integer exponents, and that's certainly not dense.

  • $\begingroup$ @T.Bongers $a_n/b_m=2^{n-m}$ is a power of $2$ for all $m$ and $n$. $\endgroup$ – Andreas Blass Aug 28 '13 at 23:34
  • $\begingroup$ Ah, yes. I misread it, sorry. $\endgroup$ – user61527 Aug 28 '13 at 23:36

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