0
$\begingroup$

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 $S_k = \{x\in\Bbb R: 0\le x<1/k\}$.

  1. Prove that, for any $k>0$, $0\in S_k$.
  2. 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.

  • 0
    Does my formatting fit with your original intention?2012-10-24
  • 0
    Yes, thanks a lot.2012-10-24

2 Answers 2