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:
$$]a,b[ \subseteq \bigcup_{n=1}^\infty \left] -\frac1{n}, 1+\frac1{n}\right[\qquad ?$$