Union and intersection of the family of sets $[-1,1-\frac1n]$; describe and prove! 
Problem: Describe $\bigcap_{n=1}^{\infty}[-1,1-\frac{1}{n}]$ and
  $\bigcup_{n=1}^{\infty}[-1,1-\frac{1}{n}]$. And prove that their results are true.

Let $A_{n}=\left [-1,1-\frac{1}{n}  \right ]$. We see that 
\begin{equation*}
A_{1}=\left [-1,0\right ]\qquad A_{2}=\left [-1,\frac{3}{4}\right ]\qquad \dots\qquad A_{\infty}=\left [-1,1\right ]
\end{equation*}
so we have
\begin{equation*}
\bigcap_{n=1}^{\infty}A_{n}=A_{1}\cap A_{2}\cap A_{3}\cap \dots\cap A_{\infty}=\left [ -1,0 \right ]
\end{equation*}
and
\begin{equation*}
\bigcup_{n=1}^{\infty}A_{n}=A_{1}\cup A_{2}\cup A_{3}\cup \dots\cup A_{\infty}=\left [ -1,1\right]
\end{equation*}
If I try to google something about it, the second one should have been $\bigcup_{n=1}^{\infty}A_{n}=[-1,1[$. It doesn't really make sense why. If it's true, then $A_{\infty}$ must be wrong. Could  you please elaborate? 
And how do I prove them more precisely by showing two different ways: $\subseteq $ and $\supseteq $? I've tried to look for some inspirations in See page 73, Example 3b but it doesn't really give me some ideas how to prove on my own.

There is no $A_\infty$: the notation is traditional and unfortunate. If you write 
$$
\bigcap_{n\ge1}A_{n}
$$
you'll probably will figure out better what's the set you want to compute. The variable $n$ is considered to take on natural number values and $\infty$ is not a natural number.
