Definitions for Quiz

I figured it might be handy to have a list of definitions to study from. So... here are some definitions from the book.

• A set $A \subseteq \mathbb{R}$ is bounded above if $\exists$ a number $b \in \mathbb{R}$ such that $a \leq b$ for all $ a \in A$. (from page 14)
• A real number $s$ is the least upper bound for a set $A \subseteq \mathbb{R}$ if it means the following two criteria:
  (i) $s$ is an upper bound for $A$;
  (ii) if $b$ is any upper bound for $A$, then $s \leq b$. (from page 14)
  
  
• The Axiom of Completeness: Every nonempty set of real numbers that is bounded above has a least upper bound. (from page 14)
• A set $A$ is countable if $\mathbb{N} \sim A$. (from page 24)
• Two sets $A$ and $B$ have the same cardinality if there exists $f: A \rightarrow B$ that is 1-1 and onto. In this case, we write $A \sim B$. (from page 23)

As for the nested interval property and the limit of sequence, the book versions are pretty wordy. Does anybody think using the versions Dr McClure gave us in class is safe?

The book definition for the Nested Interval Property isn't too too bad, the only difference from the definition in our notes is the notation.

For every $n$ $\in$ $\mathbb{N}$, assume we are given a closed interval $I_n$ = $[a_n, b_n]$ = {$x \in \mathbb{R}$ : $a_n \leq x \leq b_n$}

Assume also that each $I_n$ contains $I_{n+1}$. Then the resulting nested sequence of closed intervals has a nonempty intersection.

Does anyone know if the definitions discussed in class will be preferred over the book definitions? I ask because, although the definitions in my notes are basically the same as the ones Ghilman so kindly put up, they differ just enough for me to wonder if the discrepancy matters. My definitions for countability and cardinality are much more drawn out than the ones above...

I do think that using the definitions he gave in class would be safe, they are basically the same but with more quantifiers and less words. I recall that he wrote the following on the board and said to know this for the quiz:

$lim\ a_n = L$ if $ \forall \epsilon > 0, \exists N \in \mathbb{N}$ such that $\lvert a_n - L \rvert< \epsilon$ whenever $n\geq N$.

I notice that the book actually calls this the convergence of a sequence and just uses more words than quantifiers:

A sequence ($a_n$) converges to a real number $a$ if, for every positive number $\epsilon$, there exists an $N \in \mathbb{N}$ such that whenever $n \geq N$ if follows that $\lvert a_n - a\rvert < \epsilon$. (p. 39)

Which is basically the same....