MATH 361A Midterm 1

A set S is bounded if it is bounded below and bounded above.

A set S is bounded below if the set has a lower bound.

A set S is bounded above if the set has an upper bound.

$ \text{A real number} \thinspace l \thinspace \text{is a lower bound of the set S} \in \mathbb{R} \thinspace \text{if} \thinspace l \leq x \thinspace \text{for all} \thinspace x \in S. $

$\text{A real number} \thinspace u \thinspace \text{is an upper bound of the set S} \in \mathbb{R} \thinspace \text{if} \thinspace u \geq x \thinspace \text{for all} \thinspace x \in S. $

$ \text{A sequence is a function whose domain is the set of all natural numbers} \thinspace \mathbb{N}. $

$ m \in \mathbb{R} \thinspace \text{is called the greatest lower bound of S (AKA infimum of S [we write m = inf(S)])} $
$\text{provided m is a lower bound and} \thinspace m \geq l \thinspace \text{for all} \thinspace l \in L_s \text{(lower bounds of S)}. $

$ M \in \mathbb{R} \thinspace \text{is called the least upper bound of S (AKA infimum of S [we write M = sup(S)])}.$
$\text{provided M is an upper bound and} \thinspace M \leq u \thinspace \text{for all} \thinspace u \in U_s \text{(upper bounds of S)}.$

Since every Cauchy sequence converges, then every Cauchy sequence is bounded.

$|\text{a+b}| \leq |\text{a}|+|\text{b}|$

$\text{A sequence is monotone decreasing if} \thinspace a_\text{n} \geq a_\text{n+1}.$

$\text{A sequence is strictly decreasing if} \thinspace a_\text{n} >a_\text{n+1}$

$\text{A sequence is monotone increasing if} \thinspace a_\text{n} \leq a_\text{n+1}$

$\text{A sequence is strictly increasing if} \thinspace a_\text{n} < a_\text{n+1}$

$\text{A sequence} \thinspace \{b_\text{k}\} \thinspace \text{is called a subsequence of the sequence} \thinspace \{a_\text{n}\} \text{if there are natural numbers}$
$n_1 < n_2<n_3,...\thinspace \text{such that} \thinspace b_\text{k} = a_{\text{n}_\text{k}} \thinspace \text{for} \thinspace k = 1,2,3,... \thinspace \text{or for k} \in \mathbb{N}. $

$\text{The limit of the sequence} \thinspace \{a_\text{n}\} \thinspace \text{is written} \thinspace \lim_{\text{n}\rightarrow \infty } a_\text{n}=A$
$\forall \epsilon > 0, \exists \text{N}\in \mathbb{N} \thinspace \text{such that for all} \thinspace \text{n>N}, |a_\text{n}-A| < \epsilon.$
$\text{If} \thinspace \{a_\text{n}\} \thinspace \text{has a limit A, we say that the sequence converges; otherwise we say that} \thinspace \{a_\text{n}\} \thinspace \text{diverges}.$

$\text{A sequence is called a Cauchy Sequence if} \thinspace \forall \epsilon > 0, \exists \text{N} \in \mathbb{N} \thinspace \text{such that for all} \thinspace \text{m,n>N}, |a_\text{m}-a_\text{n}| < \epsilon. $

$\text{Let} \thinspace \text{x} \in \mathbb{R} \thinspace \text{such that} \thinspace \text{x} \geq -1 \thinspace \text{then} \thinspace (1+\text{x})^\text{n} \geq 1+\text{nx}.$

$\text{If a,b} \in \mathbb{R} \thinspace \text{with} \thinspace \text{a}>0 \thinspace \text{and} \thinspace \text{b}>0 \thinspace \text{then} \thinspace \exists \text{n} \in \mathbb{N} \thinspace \text{with} \thinspace \text{na}>\text{b}.$

Every open interval (a,b) contains a rational number.

Every interval I contains an irrational number

$\forall \epsilon>0, \exists \text{n} \in \mathbb{N} \thinspace \text{such that} \thinspace 1/\text{n} < \epsilon $

$\text{If} \thinspace \lim_{\text{n}\rightarrow \infty} a_\text{n} \thinspace \text{exists, then it is unique}.$

$\forall \text{n} \in \mathbb{N}, a_\text{n} \leq b_\text{n} \leq c_\text{n} \thinspace \text{and} \thinspace \lim_{\text{n}\rightarrow \infty } a_\text{n}=\thinspace \text{A}=\lim_{\text{n}\rightarrow \infty} c_\text{n} \thinspace \text{then} \thinspace \lim_{\text{n}\rightarrow \infty} b_\text{n}=\thinspace \text{A}$

$ (a) \lim_{\text{n}\rightarrow \infty } a_\text{n}+b_\text{n}=\text{A+B} $
$ (b) \lim_{\text{n}\rightarrow \infty } ka_\text{n}=\text{kA} $
$ (c) \lim_{\text{n}\rightarrow \infty } a_\text{n}b_\text{n}=\text{AB} $
$ (d) \lim_{\text{n}\rightarrow \infty } \frac{1}{b_\text{n}} = \text{1/B} $
$ (e) \lim_{\text{n}\rightarrow \infty } \frac{a_\text{n}}{b_\text{n}} = \text{A/B} $

$\text{If} \thinspace \{a_{\text{n}}\} \thinspace \text{is monotone increasing and bounded above then} \thinspace \{a_{\text{n}}\} \thinspace \text{converges and} \thinspace \lim_{\text{n}\rightarrow \infty } a_{\text{n}}=\text{sup}_{\text{n}}\{ a_{\text{n}}\} $

$\text{Suppose} \thinspace I_1 = [a_1,b_1],I_2=[a_2,b_2],I_3=[a_3,b_3] \thinspace \text{and} \thinspace \lim_{\text{n}\rightarrow \infty } (b_{\text{n}}-a_{\text{n}})=0$
$\text{Then there is exactly one real number common to all intervals I}$

Every bounded sequence has a convergent subsequence.

$\text{If l = lim} \thinspace \text{inf}_{\text{n}}\{a_\text{n}\} \thinspace \text{and L = lim}\thinspace \text{sup}_{\text{n}}\{a_{\text{n}}\} \thinspace \text{then} \thinspace \forall \epsilon>0, \exists \text{N} \in \mathbb{N} \thinspace \text{such that for all} \thinspace \text{n>N},l-\epsilon<a_\text{n}<L+\epsilon.$

$\text{inf C} = \lim \text{inf}_{\text{n}}\left\{a_\text{n}\right\} \thinspace \text{and} \thinspace \text{sup C}=\lim \text{sup}_{\text{n}}\left\{a_\text{n}\right\} $

$\text{Let} \thinspace \{a_{\text{n}}\} \thinspace \text{be a sequence x} \in \mathbb{R} \thinspace \text{is a cluster point of} \thinspace \{a_{\text{n}}\}\thinspace \text{if some subsequence of} \{a_{\text{n}}\} \thinspace \text{converges to x}.$

$\text{A sequence diverges to}\thinspace -\infty(\{a_\text{n}\}\rightarrow -\infty) \iff \forall \text{M}>0, \exists \text{N} \in \mathbb{N} \thinspace \text{such that} \thinspace \forall \text{n} \in \mathbb{N},a_\text{n}<-\text{M}.$

$\text{A sequence diverges to}\thinspace \infty(\{a_\text{n}\}\rightarrow \infty) \iff \forall \text{M}>0, \exists \text{N} \in \mathbb{N} \thinspace \text{such that} \thinspace \forall \text{n} \in \mathbb{N},a_\text{n}>\text{M}.$

$\lim_{\text{n} \rightarrow \infty } \text{inf}_{\text{n}} \left\{ a_{\text{n}}\right\} =-\infty $

$\lim_{\text{n} \rightarrow \infty } \text{sup}_{\text{n}}\left\{ a_{\text{n}}\right\} =\infty $

$\lim \text{inf}_{\text{n}}\left\{a_{\text{n}}\right\}=\lim_{\text{N}\rightarrow \infty}\thinspace\begin{matrix} \text{inf}\\ n>N\end{matrix}\left\{a_{\text{n}}\right\} $

$ \lim \text{sup}_{\text{n}}\left\{a_{\text{n}}\right\}=\lim_{\text{N}\rightarrow \infty}\thinspace \begin{matrix}\text{sup}\\ n>N\end{matrix}\left\{a_{\text{n}}\right\} $

$ \text{A sequence} \thinspace \{a_\text{n}\}\thinspace \text{diverges if} \thinspace \exists \epsilon>0, \forall \text{N} \in \mathbb{N}, \exists \text{n} \in \mathbb{N} \thinspace \text{such that} \thinspace |a_\text{n}-A| \geq \epsilon.$