Filtration of a set Really dont get it

Question

Let $a\in \mathbb R$. Consider the filtrations $\varphi,\varphi':\mathbb N\to {\rm Sub}(\mathbb R)$ defined by $\varphi(n)=\left(a-\frac{1}{n},a+\frac{1}{n}\right)$ and $\varphi'(n)=\left[a-\frac{1}{n},a+\frac{1}{n} \right]$ for all $n\in \mathbb N$ show that $f:\mathbb R\to \mathbb R$ is eventually constant with respect to $\varphi$ if and only if it is eventually constant with respect to $\varphi'$. Moreover show that $f:\mathbb R\to \mathbb R$ is eventually bounded with respect to $\varphi$ if and only if it is eventually bounded with respect to $\varphi'$.