Math Help - prove it!!!

1. prove it!!!

$\bigcup _ { n=1} ^ \infty (1+\frac {1}{n} , 4-\frac{1}{n})= (1,4)$

2. Hi.
$x \in \bigcup _ { n=1} ^ \infty (1+\frac {1}{n} , 4-\frac{1}{n})$ 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 $x \in (1 , 4)$.

Can you see why each 'if and only if' holds?