Proof about set Intersection

Question

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.

Answer 1

For the first part, all you need is to show that $0<\frac1k$ for all integers $k>0$. For the second part, you must show that if $x>0$, then there is some integer $k>0$ such that $\frac1k\leq x$ (hint: $\Bbb N$ has no upper bound in $\Bbb R$), then the rest of the claim follows from the first part and the definition of the $S_k$'s.