I've been stuck on this question since last night and the problem I'm having is I just don't know how to set out the answer or if I answered it correctly.
Question
Describe the following set, and prove your answer correct. (Here brackets denote intervals on $\mathbb{R}$.)
$\bigcap_{i=0}^{\infty}(\frac{-1}{i}, \frac{1}{i})$
Working
As I said I don't know how to set it out, so my answer is simply:
$() \cap (-1, 1) \cap (\frac{-1}{2}, \frac{1}{2}) \cap (\frac{-1}{3}, \frac{1}{3}) \cap ... \cap (\frac{-1}{k}, \frac{1}{k}) \cap ...$
and as for the actual working, I just kind of used my head and it seems like there isn't any working, and it says to prove my answer correct so I don't know what to do for it.
Thanks, any help is greatly appreciated! :)
We are taking the intersection of a lot of sets, so the resulting set consists of things that are in all of these sets. Another way to think about this, is if $x$ is not in a single one of the sets, then $x$ is not in the intersection.
Notice that $(-\frac{1}{i},\frac{1}{i})=\{x:|x|<\frac{1}{i}\}$. So with the intersection, we will have $\{x:|x|<\frac{1}{i}\forall i\in\mathbb{N}\}=\{0\}$. Maybe it is not obvious that this is the intersection immediately, but you should be able to recognize that the intersection is contained in this set.
Now that we have some idea of what are looking for, we start by noting $0$ is in each of the intervals given, so it must be in the intersection. Consider $x\neq 0$. Then $|x|>0$, so there is some $i\in\mathbb{N}$ such that $\frac{1}{i}<|x|$. Thus, $x\not\in(-\frac{1}{i},\frac{1}{i})$, so $x$ is not in the intersection. Thus, we have shown the only element of the intersection is $0$, so $$\bigcap_{i\in\mathbb{N}}\left(-\frac{1}{i},\frac{1}{i}\right)=\{0\}$$