Lecture - Analysis MT22, IX

Created: November 11, 2022

Flashcards

What is the scenic viewpoint theorem about any sequence $(a _ n)$?

It has a monotonic subsequence.

What set do you consider for the proof of the scenic viewpoint theorem?
\[V = \\{ k \in \mathbb{N}^{\ge 1}\space |\space \forall m > k \implies a_m < a_k \\}\]

What 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

What is the Bolzano-Weierstrauss theorem??

Any bounded real sequence $(a _ n)$ has a convergent subsequence.

Does the Bolzano-Weierstrass theorem work for complex sequences?

Yes.

What is the name of the theorem that states any bounded real sequence has a convergent subsequence?

The Bolzano-Weierstrass theorem.

What two sub-theorems are the ingredients of the Bolzano-Weierstrass theorem?
  • The scenic viewpoint theorem
  • The monotonic sequences theorem

How do you pronounce “Cauchy”?

Co-she

If $(a _ n)$ is a real or complex sequence, what does it mean to satisfy the Cauchy condition?
\[\forall \epsilon > 0 \exists N \in \mathbb{N} \text{ s.t } \forall m, n > N \; |a_n - a_m| < \epsilon\]

Why 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.

What is true about the boundedness of a sequence if it is a Cauchy sequence?

It is necessarily bounded.

What is the correspondence between convergent sequences and Cauchy sequences?

A convergent sequence is a Cauchy sequence.

What 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 $(a _ n)$?::

\[(a_n) \text{ is convergent} \implies (a_n) \text{ is a Cauchy sequence}\]


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