Analysis II HT23, Nested interval theorem
Flashcards
@State the nested interval theorem.
Let $I _ n = [a _ n, b _ n]$ be a sequence of closed intervals satisfying each of the following conditions:
- $I _ 1 \supseteq I _ 2 \supseteq I _ 3 \supseteq \ldots$
- $b _ n - a _ n \to 0$ as $n \to \infty$. Then
consists of exactly one real number $x$. Moreover both sequences $a _ n$ and $b _ n$ converge to $x$.
Proofs
https://math.gmu.edu/~dsingman/315/sect1.6nounc.pdf
@Prove the nested interval theorem:
Let $I _ n = [a _ n, b _ n]$ be a sequence of closed intervals satisfying each of the following conditions:
- $I _ 1 \supseteq I _ 2 \supseteq I _ 3 \supseteq \ldots$
- $b _ n - a _ n \to 0$ as $n \to \infty$.
Then
\[\bigcap _ {n=1}^{\infty} I _ n\]
consists of exactly one real number $x$. Moreover both sequences $a _ n$ and $b _ n$ converge to $x$.
Todo.