Suppose that for some sequence of positive random variables $N^r$ we have for any $\epsilon>0$, $P(N^r \ge \epsilon)\to 0$ as $r \to \infty$.
How can we construct a sequence $\epsilon_r \downarrow 0$ such that $P(N^r \ge \epsilon_r)\to 0$? The difficulty I have here is finding a decreasing sequence to $0$ in $r$ such that $\{N^r \ge \epsilon_r\}$ tends to zero simultaneously as $r \to \infty$. I would greatly appreciate if anyone could show me the trick in constructing such a sequence.