Proving a union of sets How to prove:
$\cup_n[\frac{1}{n},1] = (0,1]$, where $n \in \mathbb N $
The only thing I'm aware of is that we have to prove both left to right and right to left as I'm dealing with sets, and couldn't find a starting point.
Can anyone help with this?

If $x\in\cup_n[\frac{1}{n},1],$ then $x\in[\frac1n,1]$ for some $n$, so, since $\frac1n>0$, it follows that $x\in(0,1]$.
On the other hand, if $x\in(0,1]$, then $x\in[\frac1n,1]$ for all $n>\lfloor \frac1x\rfloor$, so $x\in \cup_n[\frac{1}{n},1] $.
Here $\lfloor y\rfloor$ is the floor function.
