If I show this statement:
$x\in \left] a,b \right[ \Rightarrow \exists n \in \mathbb{N} : x\in \left] -\frac1{n}, 1+\frac1{n}\right[$
Have I then shown this statement:
$]a,b[ \subseteq \bigcup_{n=1}^\infty \left] -\frac1{n}, 1+\frac1{n}\right[\qquad ?$