# Prove $\bigcap\limits_{n=1}^{\infty}(-\frac{1}{n},1+\frac{1}{n})=\left[0,1\right]$

Good night. I make a proof of this prove:

Prove $\bigcap\limits_{n=1}^{\infty}(-\frac{1}{n},1+\frac{1}{n})=\left[0,1\right]$

Proof:

Be $\left\{ x_{n}\right\} =\left\{ -\frac{1\;}{x}:\;x\;\epsilon\:\mathbb{N}\right\}$ and $\left\{ Y_{n}\right\} =\left\{ \frac{1}{y}+1\;:\;y\;\epsilon\:\mathbb{N}\right\}$ sequences such then $lim_{n\rightarrow\infty}\frac{1}{x_{n}}=0$ and $lim_{n\rightarrow\infty}\frac{1}{y_{n}}+1=1$ but when i see the intersection that confused me because the intersection is $Ø$. Can someone help me?

Hint 1: $[0,1] \subset(-\frac{1}{n},1+\frac{1}{n})$.
Hint 2: If $x \in \cap_{n=1}^{\infty}(-\frac{1}{n},1+\frac{1}{n})$ then $$-\frac{1}{n} < x < 1+\frac{1}{n}$$
What happens when you take the limit by $n$?