Imagine infinite intersection of $\bigcap_{n=1}^\infty\left(0,\,\frac1n\right)$ where $n$ is positive natural number.
Assume infinite intersection of $\left(0,\,\frac1n\right)$ is empty set. In other words, given any positive real number $p$ less than 1, there exists some $N$ such that $\frac1N$ is less than $p$. But then we have actually made one to one mapping from real numbers to rational numbers. But such a mapping is impossible for the difference of cardinality between real and rational numbers. So the infinite intersection of $\left(0,\,\frac1n\right)$ is not empty set. What's wrong with it?