I have a question regarding multiple intersections, and we have to prove that:
$\bigcap_{n=1}^\infty (1-1/n^2, 1+1/n^2) = \{1\} $
where we are talking about the open interval mentioned. I understand why the rhs $\subset$ lhs, however for the other way around, since we are required to prove it, I was thinking about proof by contradiction, i.e. if there was an $x \in $ lhs, then for $x <$ or $> 1$, there would be a contradiction. However looking at the working out after much thought, I found that they had chosen:
For $x>1$ , choose $n = \lceil 1/\sqrt(x-1) \rceil +1 \in \mathbb N $ and then they proceeded to show why $x > 1+1/n^2 $ and that is why the case $ x>1$ is not true.
This part I didn't understand, and was hoping someone could clarify. I don't understand why that particular number was chosen and how this method itself works for proof by contradiction. If someone could also cite some places as to where I can learn a little more about this method then it would be helpful as well. Furthermore if there are any other methods that could be used for these types of questions I'd appreciate it if you can explain them to me as well.
Thanks in advance. :)
Given any $x > 1$, the author would like to show that there is some integer $n \geq 1$ such that $x > 1+1/n^{2}$. Note that $x > 1 + 1/n^{2}$ if $x-1 > 1/n^{2}$, if $n^{2} > 1/(x-1)$, and if $n > 1/\sqrt{x-1}$. The number $\lceil 1/\sqrt{x-1} \rceil$ is an integer $\geq 1$ and is $> 1/\sqrt{x-1}$. So it is intuitive to choose the number.