I am having trouble formalizing two proofs I have to make about an infinite intersection of sets.
Suppose that, for every $k\in\Bbb N$ ($k>0$), we define the set $S_k = \{x\in\Bbb R: 0\le x<1/k\}$.
- Prove that, for any $k>0$, $0\in S_k$.
- Prove that $\{0\}=\bigcap_{k>0}S_k$
For the first one, I am trying to say that, as $x$ is equal or greater than $0$ for $x=0$, and this $x\in\Bbb R$, there is always an $x$ for which that statement is true (say, $x$ being $0$).
The other one is driving me nuts. I do not have any idea on how to make it. I know it is true, because there is always an element $0$ present in all $S_k$, this follows from the reasoning in the first point.
Any help would be greatly appreciated.