Problem:
For every $n$ in $\mathbb{N}$, we consider the sets $A_{n}:=\left \{ (2n+1)\lambda :\ \lambda \in \mathbb{N}\right \}$. The question is to find $\bigcap_{n=1}^{\infty }A_{n}$, and $\bigcup_{n=1}^{\infty }A_{n}$.
For the intersection, I think it is the empty set, because for every $n$ in $\mathbb{N}$; we have $n\notin A_{n}\Rightarrow n\notin \bigcap_{n=1}^{\infty }A_{n}$. For the union, I tried the first few sets, but I can't see exactly what the union of all the sets should be. Any help is appreciated.
Best Answer
These are all the positive integers that have a factor of the shape $2n+1$. That's the collection of all positive integers that have an odd factor $\ge 3$. Every positive integer qualifies, except for the powers $1,2,4,8,\dots$ of $2$.
So our set is the complement of the set of powers of $2$.