Consider a right-continuous, monotone decreasing, non-negative function $\bar F(x)$ (its the tail of a probability distribution, but that doesn't matter). Now let \begin{equation} I_{n}=\{x : \bar F(x)\leq n^{-1}\} \end{equation} and define $\bar F^{-1}(n^{-1})=\inf \,I_{n}$. I want to prove \begin{equation} \lim_{n\rightarrow\infty} n \bar F(\bar F^{-1}(n^{-1}))=\lim_{n\rightarrow\infty} n \bar F(\inf\,I_{n})=1. \end{equation} By right continuity of $\bar F(x)$, I think \begin{equation} n \bar F(\bar F^{-1}(n^{-1})) \leq 1 \end{equation} for all $n$, but I'm struggling to see how I can get an asymptotic bound from below. Any hints appreciated.
Edit: An additional piece of information that might be useful is that $\sup\{x:\bar F(x)>0\}=\infty$