Let $F$ be a strictly increasing distribution function. For a given $\tau \in (0,1)$, suppose there exists $\epsilon_{\tau}$ such that $F(\epsilon_{\tau}) = \tau$. Considering a set of independent and identically distributed (i.i.d.) random variables $X_{1}, \cdots, X_{n}$, with their order statistics denoted by $X_{(1)}, \cdots, X_{(n)}$, and let $k = [n\tau]$ (where $[ \cdot ]$ denotes the floor function).
I got the following statement from the post https://stats.stackexchange.com/a/373530/409031. Specifically, it is stated that $$X_{(k)} \to \epsilon_\tau$$ almost surely as $n$ approaches infinity. Could someone provide a detailed explanation or proof of this form of convergence?
I have some basic understanding of order statistics and the concept of almost sure convergence. However, I am struggling to connect these concepts to establish the proof or detailed explanation of this particular scenario. Any insights or step-by-step explanations would be greatly appreciated.