Let $U_{1},U_{2},U_{3},\ldots$ be a sequence of i.i.d. uniform-[0,1] random variables. Define the lower record times as: \begin{equation*} \sigma_{n+1} =\min{\{k\mid U_{k}<U_{\sigma_{n}}\}}, \end{equation*} with $\sigma_{1}=1$. I am tasked with attempting to show that the relative reduction in the $n$th record value, \begin{equation*} W_{n}:=\frac{U_{\sigma_{n}}}{U_{\sigma_{n-1}}}\sim U[0,1]. \end{equation*}
My initial thought is that the $U_{\sigma_{n}}\sim U[0,1]$ since the $U_{i}\sim U[0,1]$. Then, if $U_{\sigma_{n}}\sim U[0,1]$ it seems that $U_{\sigma_{n-1}}\sim U[0,1]$ also. My next thought would be to find the quotient distribution of $U_{\sigma_{n}}$ and $U_{\sigma_{n-1}}$ using the fact that both $U_{\sigma_{n}}$ and $U_{\sigma_{n-1}}$ are uniform-[0,1] random variables, but this appears to lead to a dead end since it is known that the quotient distribution of two uniform-[0,1] random variables, $Y=X/Z$, has distribution function \begin{equation*} F_{Y}(y) = \begin{cases}1-\frac{1}{2y} & z>1\\\frac{y}{2}& y\in[0,1]\\ 0 & \text{otherwise} \end{cases} \end{equation*}
Frankly, I do not quite know how to really begin, so it is likely that much of what I said may not be actually true/helpful in solving the problem. Can someone please help point me in the right direction on how to go about this problem? Any hints or references are appreciated.
It might be helpful to note that this is provided as a homework exercise for a module on Poisson processes, so there might be some relation there that I am missing.