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

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?

• enumerate rationals in $[a,b]$ – mathworker21 Dec 16 '18 at 15:48
• I can't see anything but empty space in desmos. – BigbearZzz Dec 16 '18 at 15:52
• @BigbearZzz I've fixed the link, thanks for pointing out – roman Dec 16 '18 at 15:53

On the other hand, you can take the sequence:$$a,b,a,\frac12a+\frac12b,b,a,\frac23a+\frac13b,\frac13a+\frac23b,b,a,\frac34a+\frac14b,\ldots$$
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.
Its famous that {$$\sin (n)\}_{n=1}^\infty$$ is dense in $$[-1,+1]$$. So, shift to the interval of interest: $$\{a+ (b-a) \sin n\}_{n\in \mathbb{N}} \ .$$