# Lecture - Analysis MT22, IX

> Source: https://ollybritton.com/notes/uni/prelims/mt22/analysis-i/lectures/9/ · Updated: 2022-11-11 · Tags: uni, lecture

- [Course - Analysis MT22](https://ollybritton.com/notes/uni/prelims/mt22/analysis-i/)

### 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 $V = \\{ k \in \mathbb{N}^{\ge 1}\space |\space \forall m > k \implies a_m < a_k \\}$ 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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