$\bigcup _ { n=1} ^ \infty (1+\frac {1}{n} , 4-\frac{1}{n})= (1,4)$
$x \in \bigcup _ { n=1} ^ \infty (1+\frac {1}{n} , 4-\frac{1}{n})$ if and only if $\left( \exists n \in \mathbb{N}\right) x \in (1+\frac {1}{n} , 4-\frac{1}{n})$.
$\left( \exists n \in \mathbb{N}\right) x \in (1+\frac {1}{n} , 4-\frac{1}{n})$ if and only if $x \in (1 , 4)$.