$ A_n $ =$[\frac{n}{n+1},\frac{n+1}{n+2}] $ be closed subsets find $\bigcup_{n=1}^\infty A_n $ Let $ A_n  =\frac{n}{n+1},\frac{n+1}{n+2}] $, $n=1,2,3...$ be closed subsets of real line R. Then $\bigcup_{n=1}^\infty A_n $ is 


*

*(1/2,1)

*[1/2,1)

*(1/2,1]

*[1/2,1]
My attempt : think it could be [1/2,1) since $\lim \frac{n+1}{n+2}$=1 . I am not sureit is just a guess.any hints plz.

Hint: Since $f(x)=\frac{x}{x+1}$ is monotonically increasing, we have that for any $0\le x\lt1$,
$$
n=\left\lfloor\frac{x}{1-x}\right\rfloor\iff n\le\frac{x}{1-x}\lt n+1\iff\frac{n}{n+1}\le x\lt\frac{n+1}{n+2}
$$
Thus, for any $\frac12\le x\lt1$, if $n=\left\lfloor\frac{x}{1-x}\right\rfloor$, then $n\ge1$ and $x\in\left[\frac{n}{n+1},\frac{n+1}{n+2}\right]$.
Furthermore, note that for all $n\ge0$, $\frac{n+1}{n+2}\lt1$.
