# Thread: Sequences

1. ## Sequences

For $n \in {\mathbb {N}}$, define $I_n$ = $[{a_n, b_n}]$ to be a sequence of closed and bounded intervals such that $I_1\supseteq {I_2} \supseteq {I_3} ...$

Prove that there is a real number $x$ such that $x \in I_n$ for all $n$ i.e.

${^\infty}$
$\bigcap {I_n}{\neq}{\oslash}$
$_{i=1}$

2. Originally Posted by yusukered07
For $n \in {\mathbb {N}}$, define $I_n$ = $[{a_n, b_n}]$ to be a sequence of closed and bounded intervals such that $I_1\supseteq {I_2} \supseteq {I_3} ...$

Prove that there is a real number $x$ such that $x \in I_n$ for all $n$ i.e.

${^\infty}$
$\bigcap {I_n}{\neq}{\oslash}$
$_{i=1}$
This is a sequence of closed subsets of $[a_1,b_1]$ which have the FIP, apply $[a_1,b_1]$'s compactness.

3. Originally Posted by yusukered07
For $n \in {\mathbb {N}}$, define $I_n$ = $[{a_n, b_n}]$ to be a sequence of closed and bounded intervals such that $I_1\supseteq {I_2} \supseteq {I_3} ...$

Prove that there is a real number $x$ such that $x \in I_n$ for all $n$ i.e.

${^\infty}$
$\bigcap {I_n}{\neq}{\oslash}$
$_{i=1}$
Since $\{a_{n}\}_{n\in\mathbb{N}}$ is nondecreasing and $\{b_{n}\}_{n\in\mathbb{N}}$ is increasing,
we have $a:=\lim\nolimits_{n\to\infty}a_{n}$ and $b:=\lim\nolimits_{n\to\infty}b_{n}$.
Then $(a+b)/2\in[a,b]=\cap_{n\in\mathbb{N}}[a_{n},b_{n}]$.