Lecture - Analysis MT22, IX
Flashcards
ObsSsbdCQlsTpNL6XgiHOc2zWhat is the scenic viewpoint theorem about any sequence $(a _ n)$?
It has a monotonic subsequence.
ObscF7iY3NE6yiNFXkLqKzJ9What set do you consider for the proof of the scenic viewpoint theorem?
ObsZT6g06NcIoSH9duFDPjp3What are the two cases for $V = \{ k \in \mathbb{N}^{\ge 1}\space \vert \space \forall m > k \implies a _ m < a _ k \}$ in the scenic viewpoint theorem?
- It is finite
- It is infinite
ObsV25BXvV7ByMLQwO0HESCIWhat is the Bolzano-Weierstrauss theorem??
Any bounded real sequence $(a _ n)$ has a convergent subsequence.
ObstdzSCuMwgrEII9qMmjjbnWhat is the name of the theorem that states any bounded real sequence has a convergent subsequence?
The Bolzano-Weierstrass theorem.
ObsrJnJrE9maHC1atEfoZoKHWhat two sub-theorems are the ingredients of the Bolzano-Weierstrass theorem?
- The scenic viewpoint theorem
- The monotonic sequences theorem
ObsC5NTeHv4wUDGztj1oly2eIf $(a _ n)$ is a real or complex sequence, what does it mean to satisfy the Cauchy condition?
Obsn3OxF7l0PkuCv7mEKAKH4Why is it nice to think about the Cauchy condition rather than the definition of convergence to a limit?
It lets you reason about the sequence without knowing the limit directly.
Obsd3xrBSnW0YKcncmX8EBfBWhat is true about the boundedness of a sequence if it is a Cauchy sequence?
It is necessarily bounded.
ObsQTaOMQI8DYYTUHc8R3oVGWhat is the correspondence between convergent sequences and Cauchy sequences?
A convergent sequence is a Cauchy sequence.
ObsvaCsZEBiKoKfcM5RfmfL3What is true about $(a _ n)$ if it is Cauchy and has a convergent subsequence?
Then $a _ n$ tends to the same limit as the convergent subsequence.
What is the Cauchy convergence criterion about $(a _ n)$?::
\[(a_n) \text{ is convergent} \implies (a_n) \text{ is a Cauchy sequence}\]