Omni-MATH / Let p₁,p₂,p₃,... be the prime numbers listed in increasing order, and let x₀ be a real number between 0 and 1 . For positive integer k , define …

Let p₁,p₂,p₃,... be the prime numbers listed in increasing order, and let x₀ be a real number between 0 and 1 . For positive integer k , define
$x_{k}=\begin{cases}0&\text{ if }x_{k-1}=0\\ \left\{\frac{p_{k}}{x_{k-1}}\right\}&\text{ if }x_{k-1}\ne0\end{cases}$ 
where {x} denotes the fractional part of x . (The fractional part of x is given by x−⌊x⌋ where ⌊x⌋ is the greatest integer less than or equal to x .) Find, with proof, all x₀ satisfying 0<x₀<1 for which the sequence x₀,x₁,x₂,... eventually becomes 0 .

Let $p_1,p_2,p_3,...$ be the prime numbers listed in increasing order, and let $x_0$ be a real number between $0$ and $1$ . For positive integer $k$ , define
$x_{k}=\begin{cases}0&\text{ if }x_{k-1}=0\\ \left\{\frac{p_{k}}{x_{k-1}}\right\}&\text{ if }x_{k-1}\ne0\end{cases}$ 
where $\{x\}$ denotes the fractional part of $x$ . (The fractional part of $x$ is given by $x-\lfloor{x}\rfloor$ where $\lfloor{x}\rfloor$ is the greatest integer less than or equal to $x$ .) Find, with proof, all $x_0$ satisfying $0<x_0<1$ for which the sequence $x_0,x_1,x_2,...$ eventually becomes $0$ .