Analysis I MT22, Monotone sequence theorem
Flashcards
What does the monotone sequence theorem say about an increasing $(a _ n)$?
It either diverges to $\infty$ or $a _ n \to \sup (\\{a _ n : n \in \mathbb{N}\\})$.
What does the monotone sequence theorem say about a decreasing $(a _ n)$?
It either diverges to $\infty$ or $a _ n \to \inf(\\{a _ n : n \in \mathbb{N}\\})$.
Given an increasing and bounded sequence $(a _ n)$, what do you show is the limit by using the suprema approximation property?
\[\alpha = \sup(\\{a _ n : n \in N \\})\]
Given a decreasing and bounded sequence $(b _ n)$, what do you show is the limit by using the infimum approximation property?
\[\alpha = \inf(\\{a _ n : n \in \mathbb{N}\\})\]