Family of sets proof with archimedean principle

Question

How to proceed with this proof

Let $A=\left\{A_{i}\right\}_{i\in I}$ a family of sets in $\mathbb{R}$ such that verifies the following properties:

1. $\forall a\in \mathbb{R}, \; (a,+\infty ) \in A$
2. $\forall i\in I, \; A_{i}^{c} \in A$
3. $\forall J\subseteq I, \; \bigcup_\limits{j\in J} A_{j} \in A$

Show that $\forall b\in \mathbb{R} \left[b, +\infty \right ) = \bigcap_\limits{n \in \mathbb{N}} (b-\frac{1}{n}, +\infty)$

I want to use the archimedean principle, but in the left side the set includes $b$ and in the right side the set is open and it does not include $b$. How do I proceed?

Thanks in advance.

Answer 1

If x is in left hand side side for all n, then x in (b - 1/n, oo).
So x is in the right hand side.

Assume x is in right hand side.
If x < b, then 0 < b - x; exists n in N with 1/n < b - x.
As that would force x out of the right hand side, b <= x.
Thus x is in the left hand side.

Low and behold, an infinite intersection of open sets can not only be closed but also not open.

Why the stuff about A? Notice that property 1 implies property 3.