# Lecture - Analysis MT22, VIII

> Source: https://ollybritton.com/notes/uni/prelims/mt22/analysis-i/lectures/8/ · Updated: 2022-11-10 · Tags: uni, lecture

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

### Flashcards
Suppose $\exists k \in \mathbb{N} \text{ s.t. } a_n > 0$ for $n \ge k$. What is true if and only if $\frac{1}{a_n} \to \infty$ as $n \to \infty$?::
$$
a_n \to 0
$$

Suppose $\exists k \in \mathbb{N} \text{ s.t. } n > k \implies a_n = -b_n$ for $n \ge k$. What is true if and only if $b_n \to \infty$?::
$$
a_n \to \infty
$$

What is the technical definition of $a_n = O(b_n)$?::
$$
\exists c > 0 \text{ and } k \in \mathbb{N} \text{ s.t. } n > k \implies |a_n| \le c|b_n|
$$

What’s the technical definition of $a_n = o(b_n)$?::
$$
\exists k \in N \text{ s.t. } n > k \implies b_n \ne 0 \text{ and } \frac{a_n}{b_n} \to 0 \text{ as } n \to \infty
$$

What does it mean for a sequence $(a_n)$ to be monotonic increasing?::
$$
\forall n > 1: a_n \le a_{n+1}
$$

What does it mean for a sequence $(a_n)$ to be monotonic decreasing?::
$$
\forall n > 1: a_n \ge a_{n+1}
$$

What does it mean for a sequence $(a_n)$ to be monotonic?::
If it is monotonic increasing or monotonic decreasing.

What is the monotonic sequence thoerem about an monotonic increasing sequence $(a_n)$?::
$$
(a_n) \text{ is convergent} \iff (a_n) \text{ is bounded above}
$$

What key idea is there that you use to prove the monotonic sequence theorem?::
Using the completeness axiom.

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