<s>[INST] Limit Set of All Real Numbers Not even sure how to start this one. Does anyone know how to do this?
Prove that there exists a sequence such that its limit set is the set
of all real numbers
Limit set is the set of all subsequence limits btw
 [/INST] Start the sequence with $0$. 
Then write $-1,1$.
Then write $-\frac{4}{2},-\frac{3}{2},-\frac{1}{2}, \frac{1}{2},\frac{2}{2},\frac{3}{2},\frac{4}{2}$. 
Then write $-\frac{16}{4},-\frac{15}{4}, -\frac{13}{4},\dots, -\frac{1}{4},\frac{1}{4},\dots, \frac{16}{4}$. 
Then write $-\frac{64}{8}, \frac{63}{8},\dots, \frac{63}{8},\frac{64}{8}$, skipping $\frac{0}{8}$, though it really doesn't matter.
Then write everything from $-\frac{256}{16}$ to $\frac{256}{16}$, in steps of $\frac{1}{16}$, skipping \frac{0}{16}$.  
Continue. 
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