I need help for the following task:
a)Formulate the Bolzano-Weierstrass theorem: Each bounded sequence in $\mathbb{R^n}$ has a convergent subsequence.
b) Let {$a_n$}$_{n\in\mathbb{N}}$ be a sequence that is monotonically increasing and {$b_n$}$_{n\in\mathbb{N}}$ a sequence that is monotonically decreasing. Both are sequences of real numbers with the property $a_n$$\leq b_n$ $\forall$ $n \in \mathbb{N}$. Show that:
$i)$ $\bigcap^\infty_{n=1}$ $[a_n,b_n]$ $\neq \emptyset$
The only approach I have is: Since {$a_n$} is monotonically increasing we have $a_n\leq a_{n+1}$. And for {$b_n$} we have $b_{n+1}\leq b_n$. Since {$a_n$}$\leq${$b_n$}, are we allowed to say, that {$a_n$} is bounded? Then if yes, we could use the Bolzano-Weierstrass theorem. But in which way? How can I reach: [$a_1,b_1$]$\cap $[$a_2,b_2$]...$\cap$[$a_n,b_n$]=$\emptyset$
$i)$ If we have additionally, that $b_n - a_n$ $\rightarrow 0$ for $n \rightarrow 0$, then $x \in \mathbb{R}$ exists with
$\bigcap^\infty_{n=1}$ $[a_n,b_n]$ = {x}
Well here we have the addition, that {$b_n - a_n$} is a convergent sequence, right? But thats all I have unfortunately...
I am thankful for any advice.
