Notes - Analysis II HT23, Nested interval theorem

Created: January 30, 2023

Flashcards

Can you 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
\[\bigcap_{n=1}^{\infty} I_n\]

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 $n \to \infty$.

Thenyn

\[\bigcap_{n=1}^{\infty} I_n\]

consists of exactly one real number $x$. Moreover both sequences $a _ n$ and $b _ n$ converge to $x$.

Todo.



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