Lecture - Analysis MT22, V

What does it mean for two sets $A$, $B$ if $A \approx B$?

They are equal in size.

When, in terms of mappings, is it true for two sets $A$, $B$ that $A \approx B$?

If there exists a bijection $f: A \to B$

What does it mean for two sets $A, B$ if $A \succcurlyeq B$?

There exists an surjection $f: A \to B$.

What does it mean for two sets $A, B$ if $A \preccurlyeq B$?

There exists a injection $f: A \to B$.

What are the two types of infinite sets?

What is the formal definition of a sequence $(a _ n)$?

A function $a : \mathbb{N} \to \mathbb{R}$.

What’s the difference between the notation for sequences $(a _ n)$ and $a _ n$?

$(a _ n)$ is the sequence, and $a _ n$ is a term of the sequeucne.

What is the formal definition of a sequence $(a _ n)$ converging to $L$?

\[\forall \epsilon > 0\,, \exists N \in \mathbb{N} \text{ s.t. } \forall n\ge N, |a_n - L|<\epsilon\]

\[\forall \epsilon > 0\\, \exists N \in \mathbb{N} \text{ s.t. } \forall n\ge N, \vert a _ n - L \vert <\epsilon\]

In this definition of convergence, why is $N$ allowed to depend on $\epsilon$?

Since the quantifier comes afterwards.

What’s the basic jist of proving there exists square roots with the set $S = \{s \in \mathbb{R} : s > 0, s^2 < 2\}$?

Showing $\alpha = \sup S$ can’t be $\alpha^2 > 2$ or $\alpha^2 < 2$, so it has to be equal $\alpha^2 = 2$.