Q:

Writing a singleton as a countable intersection

Let $x \in \mathbb{R}$. Then I am almost sure, that $$\{x\} = \bigcap_{n \in \mathbb{N}}(x - 1/n,x]$$
Any ideas how to prove this?

A:

Obviously $x\in(x-\frac{1}{n},x]$ for all $n$. So $x\in\bigcap_{n \in \mathbb{N}}(x - 1/n,x]$. Also let $y\neq x$. Then we can find $n_0\in \mathbb{N}$ such that $x-y>\frac{1}{n_0}$. Then $y\not\in(x-\frac{1}{n},x]$ for all $n\geq n_0$. Hence $y\not\in\bigcap_{n \in \mathbb{N}}(x - 1/n,x]$.

