So as a homework problem I was asked to find the intersection of the following family of sets. I was told that my reasoning was invalid as I did not correctly prove that the intersection is the empty set. Is the fact that $\displaystyle \frac{1}{n} \lt \frac{1}{n+1}$ not sufficient to show that there are no common elements over the entire family of sets and thus the intersection contains no elements?
1. Find the union and intersection of each of the following families.
(g) $\displaystyle \mathscr{A} =\{A_{n} :n\in \mathbb{N}\}$, where $\displaystyle A_{n} =\left( 0,\frac{1}{n}\right)$ for each natural number $\displaystyle n$.
Solution.
Since $\displaystyle n\in \mathbb{N}$, and $\displaystyle 0$ is not in any set because it is an open interval, it follows that the quantity $\displaystyle \frac{1}{n} \leq 1$. Therefore, the union of the family of sets $\displaystyle \mathscr{A}$ is $\displaystyle \bigcup _{n\in \mathbb{N}} A_{n} =( 0,1)$.
Since for each set $\displaystyle A_{n}$ the quantity $\displaystyle \frac{1}{n} \lt \frac{1}{n+1}$, and $\displaystyle 0$ is not in any set because it is an open interval, it follows that the intersection of the family of sets is $\displaystyle \bigcap _{n\in \mathbb{N}} A_{n} =\emptyset $, since no two sets have a common element.