1
$\begingroup$

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.

$\endgroup$
  • 4
    $\begingroup$ 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
  • 1
    $\begingroup$ 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
4
$\begingroup$

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.

$\endgroup$
  • $\begingroup$ @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

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.