Slider comments on Impossibility results for unbounded utilities

[Slider]

New attempt

$X_{\infty }=\frac{1}{2}{X}_0+\frac{1}{4}{X}_1+\frac{1}{8}{X}_2+\frac{1}{16}{X}_4+\dots$

I think I now agree that $X_0$ can be written as $\frac{1}{2}{X}_0+\frac{1}{4}{X}_0+\frac{1}{8}{X}_0...$

However this uses a “de novo” indexing and gets only to

$\frac{1}{2}$ ($\frac{1}{2}{X}_0+\frac{1}{4}{X}_0+\frac{1}{8}{X}_0...$)$+\frac{1}{4}{X}_1+\frac{1}{8}{X}_2+\frac{1}{16}{X}_4+\dots$

taking terms out form the inner thing crosses term lines for the outer summation which counts as “messing with indexing” in my intuition. The suspect move just maps them out one to one

$(\frac{1}{4}{X}_0+\frac{1}{4}{X}_1)+(\frac{1}{8}{X}_0+\frac{1}{8}{X}_2)+(\frac{1}{16}{X}_0+\frac{1}{16}{X}_4)+...$

But why is this the permitted way and could I jam the terms differently in say apply to every other term

$(\frac{1}{4}{X}_0+\frac{1}{4}{X}_1)+\left(\frac{1}{8}{X}_2\right)+(\frac{1}{8}{X}_0+\frac{1}{16}{X}_4)+\frac{1}{32}{X}_8+(\frac{1}{16}{X}_0+\frac{1}{64}{X}_{1}6)+...$

If I have $\left(\sum _{i=0}^a{x}_i\right)+\left(\sum _{j=0}^a{y}_j\right)$ I am more confident that they “index at the same rate” to make $\sum _{u=0}^c{x}_u+y_u$. However if I have $\left(\sum _i^a{x}_i\right)+\left(\sum _j^b{y}_j\right)$ I need more information about the relation of a and b to make sure that mixing them plays nicely. Say in the case of b=2a then it is not okay to think only of the terms when mixing.