Lecture - Analysis MT22, VI
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 $n > N$, $ \vert a _ n - L \vert < \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, $(b _ n)$?
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 $L < M$?
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.