Given any $\displaystyle r>0$ there is a positive integer $\displaystyle K$ such that $\displaystyle K>r$.
Is this true $\displaystyle r\notin A_K?$
So?
Another way of thinking about is that $\displaystyle \left[\bigcap_{n\in\mathbb{N}}A_n\right]'=\bigcup_{n\in\mathbb{N}}A_n'=\bigcup_{n\in\mathb b{N}}(-\infty,n)$. Now I surely hope that you recognize that this is $\displaystyle \mathbb{R}$.