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I want to:

construct a sequence subsequential limits of which form a range $[a, b]\subset \Bbb R$

Not sure how to express myself correctly but I will try.

I was thinking of a range from $a$ to $b$. If i could split that range infinitely many times and always choose either left or right part, then I could use that "path" as a subsequence. From that i should be able to find infinitely many subsequences which with some reordering will form a sequence containing any number in the range $[a, b]$ as its subsequential limit.

I've been playing around with this idea in desmos, here is a basic sketch of the idea.

Could someone help me construct such a sequence? Is my approach even valid? If so then how could i finish it?

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  • $\begingroup$ enumerate rationals in $[a,b]$ $\endgroup$ – mathworker21 Dec 16 '18 at 15:48
  • $\begingroup$ I can't see anything but empty space in desmos. $\endgroup$ – BigbearZzz Dec 16 '18 at 15:52
  • $\begingroup$ @BigbearZzz I've fixed the link, thanks for pointing out $\endgroup$ – roman Dec 16 '18 at 15:53
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When I click on your link, I just get the Desmos home page.

On the other hand, you can take the sequence:$$a,b,a,\frac12a+\frac12b,b,a,\frac23a+\frac13b,\frac13a+\frac23b,b,a,\frac34a+\frac14b,\ldots$$

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  • $\begingroup$ That's the one! I just could't wrap my mind around it, thank you! $\endgroup$ – roman Dec 16 '18 at 15:57
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    $\begingroup$ I just realized that my answer is essentially the same as you. I should have read carefully first. $\endgroup$ – BigbearZzz Dec 16 '18 at 15:59
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If $[a,b]=[0,1]$ then you can take the sequence $$ \frac 11, \frac 12, \frac 22, \frac 13, \frac 23, \frac 33, \frac 14, \frac 24, \frac 34, \frac 44, \dots $$ It shouldn't be hard to see that any point in $[0,1]$ can be approximated arbitrarily well.

For general $[a,b]$ just linearly transform the above.

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Its famous that {$\sin (n)\}_{n=1}^\infty$ is dense in $[-1,+1]$. So, shift to the interval of interest: $$ \{a+ (b-a) \sin n\}_{n\in \mathbb{N}} \ .$$

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