How do I simplify the following union of sets? Am a bit confused on how a union of sets can be arrived at, am not sure how to start even and some guidance is much appreciated, below is my attempt
$$\bigcup_{n\in\mathbb{N}} \left[\frac{1}{2n},\;8-\frac{1}{n}\right)$$
so we can say that 1/2n ≤ x < 5-1/n for all nεΝ


*

*for n = 1, we have [0.5, 7)

*for n = 2, we have [0.25, 7.5)

*for n = 3, we have [0.1667, 7.6667)

*for n = 4, we have [0.125, 7.75)


as n grows, (1/2n) gets smaller and smaller and thus approaches 0, but never is equal to 0 
AND 
as n grows, (1/n) gets smaller and smaller, thus (8-(1/n)) approaches 8 but never is equal to 8
Therefore our union of sets is (0,8)
Does this make sense?
Thanks
 A: If you want to show that $(0, 8)$ is equal to your set, show that if $x \in (0, 8)$ then $x$ is in one of the sets in the union, and then show that given an element in the union of your sets, then that element is in $(0, 8)$. That's the definition of set equality.
A: Yes, that makes sense. If you want to be completely rigorous, you would prove explicitly that for any $x$ with $0<x<8$ you can find an $n$ such that $x$ is a member of the $n$th set in the union.
Also you'd need to at least assert the (fairly obvious) fact that no number $\le 0$ or $\ge 8$ can be in the union.
However, whether you need to go to such extremes depends a bit on the purpose of the exercise -- if it's just to make sure you have an intuitive feel for how these infinite unions work, then what you have now is completely fine.
And in anything but an introductory classroom situation (or writing scripts for a computer-based formal proof checker), writing down the reasoning more formally than what you already have would be wild overkill.
