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?