Proving that the intersection of a certain family is void. prove that if x,y > 0, then there exists $n\in \mathbb{N}$, such that $n > \frac{y}{x}.$ Consequently prove that the family $$\\{(0,\frac{x}{n}]:n\in \mathbb{N} \\}$$ has a void intersection.

1. $ \mathbb N$ has no upper bound. This follws from the completeness axiom in $ \mathbb R$. Hence $\frac{y}{x}$ is not an upper bound of $ \mathbb N$.



Consequence: there is $n \in \mathbb N$ with $n > \frac{y}{x}.$

2. Suppose to the contrary that the family is not void. Then there is $y>0$ such that



$y<\frac{x}{n}$ for all $n$. This gives $n < \frac{y}{x}$ for all $n$.

But this contradicts 1.