# Lecture - Analysis VI

> Source: https://ollybritton.com/notes/uni/prelims/mt22/analysis-i/lectures/6/ · Updated: 2022-11-05 · Tags: uni, lecture

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

### Flashcards
How can you think of proving if a sequence is convergent as a game, where Player A believes $L$ is the limit of a sequence?::
Player B gives them a value of $\epsilon > 0$, and they have to find an $N$ such that for all $n > N$, $|a_n - L| < \epsilon$.

What is the $k$-th tail of a sequence $(a_n)$?::
The sequence starting $a_{k+1}, a_{k+2}, a_{k+3}, \ldots$

What is the tails lemma for $(a_n)$ and its $k$-th tail, $(b_n)$?::
$$
(a_n)\text{ converges} \iff (b_n)\text{ converges}
$$

What is the sandwiching lemma for sequences $(x_n), (a_n), (y_n)$ and $x_n \le a_n \le y_n$?::
If $x_n \to L$ and $y_n \to L$ then $a_n \to L$.

What’s another name for the sandwiching lemma?::
The squeeze theorem.

What’s the “preservation of weak inequalities” property for a sequence $a_n \to L$ and $b_n \to M$?::
If for all $n\ge 1$, $a_n \le b_n$ then $L \le M$

What’s a counter example to the incorrect “preservation of strict inequalities”, i.e. if $a_n \to L$, $b_n \to M$ and for all $n \ge 1$, $a_n < b_n$ then $L < M$?::
$$
a_n = 0
$$
$$
b_n = \frac{1}{n}
$$

What can you say about the convergence of a sequence if it is bounded?::
Nothing. It’s the converse that’s always true.

What can you say about the boundedness of a sequence if it is convergent?::
A sequence is necessarily bounded if it is convergent.

What can you say about the divergence of a sequence if it is unbounded?::
A sequence is necessarily divergent if it is unbounded.

What can you say about the boundedness of a sequence if it is divergent?::
Nothing. It’s the converse that’s always true.

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