Glossary for TEE

## TEE Terms

Monotone Sequence
A sequence $(x_n)$ in $\mathbb{R}$ with $x_n\leq x_m$ for all $n<m$ (monotone increasing), or $x_n\geq x_m$ for all $n<m$ (monotone decreasing). In other words, the values in the sequence "always go up" or "always go down".

## TEE Theorems

Monotone Sequence Theorem
Every monotone, bounded sequence in $\mathbb{R}$ converges.

# WPR I Glossary

This page contains a list of terms used in the course, and will be available for use on the WPR and TEE.

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## Terms

Cartesian product
Given sets A and B, the Cartesian product of A and B is the set A x B of all ordered pairs (a, b) such that a $\in$ A and b $\in$ B.
Cauchy Convergence Criterion for Sequences
A sequence ($a_n$) in $\mathbb{R}$ converges iff $\forall$ $\varepsilon$ > 0 $\exists$ N $\in$ $\mathbb{N}$ such that n, m $\geq$ N $\Rightarrow$ |$a_n - a_m$| < $\varepsilon$.
Complete
There are two definitions. First: The set $\mathbb{R}$, constructed by means of Dedekind cuts, is complete if it satisfies the Least Upper Bound Property. Second, $\mathbb{R}$ is complete with respect to Cauchy sequences in the sense that if ($a_n$) is a sequence of real numbers obeying a Cauchy condition then it converges to a limit in $\mathbb{R}$.
Continuous function
The function $f : [a,b] \to \mathbb{R}$ is continuous if for each $\varepsilon$ > 0 and each $x \in [a, b]$ there is a $\delta$ > 0 such that $t \in [a, b]$ and $|t - x| < \delta \implies |f(t) - f(x)| < \varepsilon$.
Convergence to a limit
The sequence ($a_n$) converges to the limit b$\in$ $\mathbb{R}$ as $n\to\infty$ provided that for each $\epsilon$>0 there exists N $\in$ $\mathbb{N}$ such that for all n$\geq$N, $|a_n-b|$<$\epsilon$
Convex subset
A set $S\subset\mathbb{R}$ is convex if for any two points $x,y\in$$S, then tx + (1-t)y\in$$S$ for all $t\in$[0,1].
Cut (or Dedekind cut)
A Cut in the rationals is a pair of subsets such that
• $\ A\cup B$=$\mathbb{Q}$, where A,B are not the empty set
• $\ A\cap B$=$\o$
• If a$\in$A and b$\in$B then a<b
• A contains no largest element
Disjoint
If $A\cap B$ is the empty set, then A and B are said to be disjoint.
Equivalence relation
A relation between elements of a common set which satisfies the following three properties for $x,y,z\in S$
• $x\sim x$
• $x\sim y$ implies that $y\sim x$
• $x\sim y\sim z$ implies that $x\sim z$
Field
A field is a set of elements in which the two operations of addition and multiplication have the algebraic properties of being well-defined, natural, commutative, associative, and have additive and multiplicative inverses, respectively.
Greatest lower bound (infimum)
The most negative value that a set approaches; the highest value of everything less than the particular set.
Injection
If for each pair of distinct elements a,a'$\in$A, the elements f(a), f(a') are distinct in B. That is, a$\neq$a' $\Rightarrow$ f(a)$\neq$f(a').
Irrational number
Any number that cannot be expressed in the form a/b, where a,b$\in \mathbb{Z}$
Least upper bound (supremum)
The most positive value that a set approaches; the lowest value of everything greater than the particular set.
Least upper bound property
A nonempty subset in $\mathbb{R}$ that is bounded above has a least upper bound in $\mathbb{R}$.
Inner product
Also known as the Dot Product of two vectors. Can be written as $\langle x,y\rangle= x_1y_1+\centerdot\centerdot\centerdot +x_my_m$. $\langle x,y\rangle$ It is a field of scalars in $\mathbb{R}$. Using the Cauchy-Schwarz inequality for elements of $x,y$, we can say that: $|\langle x,y \rangle|\leq ||x||\centerdot ||y||$ with equality if and only if $x$ and $y$ are linearly dependent.
Magnitude
absolute value, or the length of a vector. Can be defined as: $|x|=\sqrt{\langle x,x\rangle}=\sqrt{x_1^2+\centerdot\centerdot\centerdot +x_m^2}$.
Norm
A norm on a vector space V is any function ll ll : V $\to$ $\mathbb{R}$ with the three properties of vector length, if v,w$\in$V and $\lambda$$\in \mathbb{R} then: a) ll v ll \geq 0 and ll v ll = 0 iff v = 0. b) ll \lambdav ll = l \lambda l ll v ll. c) ll v+w ll \leq ll v ll + ll w ll. Rational numbers denoted as \mathbb{Q}; a fraction of integers in which the denominator is \neq 0. Surjection f: A\toB is a surjection if for each b\inB there is at least one a\in}A such that f(a)=b. Transitivity x<y<z implies x<z. Translation x<y implies x+z<y+z Trichotomy This is an additional property of the cut order. It states: either x<y, y<x, or x=y, but only one of the three things is true. Unit ball A set containing all elements with a magnitude of less than one or equal to one, S=\{\vec{s}\in\mathbb{R}^3 | \: |\vec{s}|\leq 1\}. ## Theorems Unless otherwise stated, you are free to appeal to these results on the exam. Cauchy-Schwarz inequality For all x, y \in \mathbb{R}^m, \langle x, y \rangle \leq|x| |y|. Triangle inequality For all x, y \in \mathbb{R} |x + y| \leq|x| + |y|. • Also, |x + y|^2 \leq |x|^2 + 2|x||y| + |y|^2 = (|x|+|y|)^2 The \epsilon principle This is Theorem 8 in the book. There are 2 parts. First: If a,b \in\mathbb{R} and if for each \epsilon >0, a \leq b+\epsilon, then a \leq b. Second: If x,y \in\mathbb{R} and for each \epsilon >0, |x-y| \leq\epsilon, then x=y. The "interval" theorem (Theorem 7 in the text) Every interval (a,b), no matter how small, contains both rational and irrational numbers. The intermediate value theorem A continuous function f defined on an interval [a,b] takes on absolute minimum and absolute maximum values: for some x_0, x_1 \in [a,b] and for all x \in [a,b], f(x_0) \leq f(x) \leq f(x_1). Fundamental Theorem of Continuous Functions Every continuous real valued function of a real variable x\in[a,b] is bounded, achieves minimum, intermediate, and maximum values, and is uniformly continuous. ## General To show a function is continuous you have to … Start with "Given x an element of the domain, let \varepsilon > 0." Then, solve |f(x) - f(y)| < \varepsilon for |x - y|, which determines \delta in terms of \varepsilon. Then, state "Let y be chosen such that |x - y| < \delta (substitute the \delta you calculated)." Finally, solve it for \varepsilon to prove that |f(x) - f(y)| < \varepsilon. To show a function is not continuous you have to … Show that an \epsilon>0 exists such that for all \delta >0 there exists a u with |x-u|< \delta but |f(x)-f(u)|>\epsilon To show a set is convex you have to … Show that any convex combination of the vectors \mathbf{w_1},…, \mathbf{w_k}$$\in$$\mathbb{R}, \mathbf{w}=s_1$$\mathbf{x_1}$+…+$s_k$$\mathbf{x_k}, such that s_1+\cdots+s_k=1 and 0\leq$$s_1$,…$\leq$$s_k$$\leq$1 is in the set.
To show a set is not convex you have to …
Show there exists a convex combination (as described above) such that this convex combination is not an element of the set.

# WPR II Glossary

## Terms (Sequences)

Bounded sequence
A sequence is bounded if the set consisting of the sequence is bounded (see below for the definition of a bounded set).
Cauchy criterion
A sequence $(a_n)$ in $\mathbb{R}$ such that $\forall \epsilon >0$ $\exists N \in \mathbb{N}$ such that $n, m \geq N \Rightarrow |a_n - a_m| < \epsilon$.
Convergence (in metric)
A sequence $(a_n)$ in metric space $X$ converges to $x \in X$ if for all $\epsilon > 0$ there exists $N \in \mathbb{N}$ such that $d(a_n, x) < \epsilon$ for all $n > N$.
Diverge
A sequence diverges if it does not converge to a limit p.
Sequence
A sequence of points in a metric space M is a list $p_1,p_2,\ldots$ where the points $p_n$ belong to M.
Subsequence
Just like a set can have a subset, a sequence can have a subsequence. Ex. the sequence $2,4,6,8,...$ is a subsequence of $1,2,3,4,...$. Every subseqence of a convergent sequence converges to the same limit as does the mother sequence.

## Terms (Topology)

Boundary
$\partial S = \overline{S} \cap \overline{S^{c}}$ (Closure minus the interior)
Clopen
Subset of the set M that are both closed and open. Fact: In $\mathbb{R}$ the only clopen sets are $\varnothing$ and $\mathbb{R}$ .
Closed
A set S is closed if it contains all its limits.
Closure
$\overline{S} = \bigcap K$ where $K$ ranges through the collection of all closed sets that contain $S$. Equivalently, $\overline{S}$ = {$x \in M$: if K is closed and $S \subset K$ then $x \in K$}.
Complete metric space
A metric space M is complete if each Cauchy sequence in M converges to a limit in M.
Infinite Intersection
An infinite intersection of closed sets is always closed, while the same cannot be said for open sets.
Infinite Union
An infinite union of open sets is always open, but the infinite union of closed sets is not always closed.
Interior
int($S$) = $\bigcup U$ when $U$ ranges through the collection of all open sets contained in $S$. Equivalently, int $S$ = {$x \in M$: for some open $U \subset S, x \in U$}.
Limit point (of a set)
A point $p\in M$ is called a limit point of a set, $S$ if there exists a sequence ($p_n$) in $S$ that converges to it.
Metric space
A set M, the elements of which are referred to as points of M, together with the metric d having the three properties that distance has in Euclidean Space. The three distance properties are: for all $x, y, z \in M$
• a.) positive definiteness: $d(x,y) \geq 0$, and $d(x,y) = 0$ if and only if $x = y$.
• b.) symmetry: $d(x,y) = d(y,x)$.
• c.) triangle inequality: $d(x,z) \leq d(x,y) + d(y,z)$.
Neighborhood
A neighborhood of a point p in a metric space M is any open set V that contains p.
Open
A set S is open if for each p$\in$S there exists an r > 0 such that d(p,q) < r $\Rightarrow$ q$\in$S.
Proper
Let $A$ be a subset of $M$, a metric space. If $A$ is neither the empty set nor is it $M$, then $A$ is a proper subset of $M$.
Topology
The study of "how close points are." 2 points are "close" if they are usually in the same open set. There are two important facts about topology:
• Fact 1: Any union of open sets is open. A finite intersection of open sets is open
• Fact 2: Any intersection (finite or infinite) of closed sets is close. A finite union of closed sets is closed.

## Terms (Important Properties of Sets)

Bounded set
A subset $S$ of a metric space $M$ is bounded if for some $p \in$ $M$ and some $r>0$, $S \subset M_r p$. Basically, it cannot have an infinitely large size.
Compact (sequential definition)
A set $S$ is compact if every sequence $(a_n)$ of $S$ has a convergent subsequence $(a_n_k) \to a$, such that $a \in S$.
Compact (covering definition)
A set $S$ is covering compact if every open cover ${U_\alpha}$ has a finite subcover.
Connected
A set M is connected if it is not disconnected—-that is it contains no proper clopen subset.
Disconnected
If M has a proper clopen subset A, M is disconnected. For there is a separation of M into proper, disjoint clopen subsets, M = A $\sqcup$ AC
Separation
Occurs between 2 disconnected subsets.

## Terms (Functions)

Continuous function (epsilon-delta definition)
A function $f : M\to N$ is considered continuous if it satisfies the following condition: $\forall \epsilon$ > $0$, there exists $\delta>0$ such that $q\in M$ and $d(p,q)$ < $\delta \Rightarrow d(f_p,f_q)$ < $\epsilon.$
Continuous function (convergence of limit definition)
A function $f : M\to N$ is considered continuous if and only if it sends each convergent sequence in M to a convergent sequence in N. $\lim_{n\to 0} f(x_n)=f(x)$ for all sequences $x_n$ which converge to $x$.
Continuous function (open/closed set definition)
A function $f : M\to N$ is considered continuous if it satisfies any of the following conditions:
1. The $\epsilon, \delta$ definition
2. The inverse image of any closed set in N is closed in M.
3. The inverse image of any open set in N is open in M.
Image
The image of a function, $f$, is the subset of the target, {$b\in B:$ there exists at least one element $a\in A$ with $f(a)=b$}. Note that the image is also commonly called the range of a function.
Inverse image
Let $f:M\to N$ be given. The pre-image (inverse image) of a set $V \subset N$ is $f^{pre}(V)=\{p \in M : f(p) \in V \}$.

## Theorems

Bolzano-Weierstrass Theorem
Any bounded sequence in $\mathbb{R}^m$ has a convergent subsequence.
Continuity and Convergence Theorem (Theorem 2 in the text)
A function $f$ is continuous if the sequence $(a_n)$ converges implies that the function $f(a_n)$ converges. For example, $f(\lim_{n\to \infty} a_n) = \lim_{n\to \infty} f(a_n)$.
Extreme Values Theorem
The image f(A) of continuous function f on a compact domain A is compact
Heine-Borel Theorem
Every closed and bounded set in $\mathbb{R}^n$ is compact.
Intermediate Value Theorem (generalized)
If $f$ is a real-valued function on a connected domain A, then $f$ has the intermediate value property.
Subsequence Convergence Theorem (Theorem 1 in the text)
Every subsequence of a convergent sequence converges and it converges to the same limit as does the mother sequence.

## General

How to prove that a sequence converges
A sequence is a function $f:\mathbb{N} \to M$. The $n^t^h$ term in the sequence is $f(n)=p_n$.

To prove that the sequence $(p_n)$ converges to the limit $p$ in $M$:
First, Let $\epsilon>0$. Then, you want to pick an $N \in\mathbb{N}$ such that if $n \in\mathbb{N}$ and $n \geq N$, then $d(p_n, p)<\epsilon$.

Ex: Prove that $a_n=\frac{1}{n}$ converges to zero.
Proof: Let $\epsilon>0$.
(We want to show that $\exists N \in\mathbb{N}$ such that if $n \geq N$, then $|\frac{1}{n}-0|<\epsilon$.)
Let $N>\frac{1}{\epsilon}$
If $n\geq N$, then $|\frac{1}{n}-0|=\frac{1}{n}<\epsilon$ because
$\frac{1}{n}<\epsilon\Leftrightarrow$$1<\epsilon(n)$$\Leftrightarrow\frac{1}{\epsilon}<N \leq n$. $\blacksquare$

How to prove that a sequence doesn't converge
To show that a sequence doesn't converge, one must show that there exists an $\epsilon>0$, such that for all integers N, there exists an integer n such that n>N and $d(a_n,a_N)\geq\epsilon$.

Ex: Prove that $a_m=m$ does not converge.

Proof: We want to show there exists an $\epsilon$ such that for each integer N, there exists an integer n such that n>N and $d(a_n,a_N)\geq\epsilon$.
For $\epsilon=\frac{1}{2}$, any positive integer N, and the integer n=N+1, $d(a_n,a_N)=|(N+1)-N|=1\geq\frac{1}{2}=\epsilon$. This completes the proof. $\blacksquare$

How to prove a set is compact
If a set is a subset of $\mathbb{R}^m$, then it is sufficient to show that the set is closed and bounded to prove that it is combact (by the Heine-Borel Theorem). If the set is not a subset of $\mathbb{R}^m$, then one must show that every sequence $a_n$ in the set has a subsequence $a_n_k$ that converges to a limit in the set.
How to prove a set is connected
One must show that the set contains no proper clopen subsets.
How to prove a set is disconnected
To prove a set is disconnected, one must show that the set can be seperated into proper, disjoint clopen subsets.

# WPR III Glossary

## Terms/Theorems (Differentiation)

Differentiable Function
The function $f: (a,b)\to \mathbb{R}$ is differentiable at x if $\lim_{t\to x} \frac{f(t)-f(x)}{t-x}=L$ exists. Be definition this means $L$ is a real number and for each $\epsilon > 0$ $\exists$ $\delta > 0$ such that if $0 < \mid t-x \mid < \delta$ then the differential quotient above differs from $L$ by $< \epsilon$.
The Rules of Differentiation

(a) Differentiablility implies continuity.
(b) If $f$ and $g$ are differentiable at $x$ then so is $f + g$, the derivative being $(f + g)'(x) = f'(x) + g'(x)$.
(c) If $f$ and $g$ are differentiable at $x$ then so is $f \cdot g$, the derivative being $(f \cdot g)'(x) = f'(x) \cdot g(x) + f(x) \cdot g'(x)$.
(d) The derivative of a constant is zero, $c' = 0$.
(e) If $f$ and $g$ are differentiable at $x$ and $g(x) \neq 0$ then $f / g$ is differentiable at $x$, the derivative being $(f / g)'(x) = \frac{(f'(x) \cdot g(x) - f(x) \cdot g'(x))}{g(x)^2}$.
(f) If $f$ is differentiable at $x$ and $g$ is differentiable at $y = f(x)$ then $g \circ f$ is differentiable at $x$, the derivative being $(g \circ f)'(x) = g'(y) f'(x)$.

Mean Value Theorem
A continuous function $f: [a,b]\to \mathbb{R}$ that is differentiable on the interval (a,b) has the mean value property: there exists a point c$\in$(a,b) such that f(b)-f(a)=f'(c)(b-a).
Smoothness Classes
If $f$ is differentiable and its derivative function $f'$(x) is a continuous function of x, then $f$ is continuously differentiable, and $f$ is of class $C^1$. If $f$ is $r^{th}$ order differentiable and $f^{(r)}$(x) is a continuous function of x, then $f$ is continuously $r^{th}$ order differentiable and $f$ is of class $C$r. If $f$ is smooth, then it is of class $C$r for all finite $r$ and we say that $f$ is of class $C^\infty$. A continuous function is of class $C^0$.
Uniform Continuity
A function $f$ is uniformly continuous on an interval [a,b] or (a,b) if for each $\epsilon > 0$ $\exists$ $\delta > 0$ such that if $0 < \mid x-t \mid < \delta$, then $|f(x)-f(t)|<\epsilon$. Note that $\delta$ can only depend on $\epsilon$ and not x, different from regular continuity where $\delta$ can depend on both x and $\epsilon$.

## Terms/Theorems (Series)

Absolute Convergence
If the series converges absolutely, then every rearrangement converges to the same value. An example is: $\sum a_n$ converges absolutely if $\sum |a_n|$ converges.
Comparison Test
If $\sum b_n$ converges to $\beta$, and $|a_n| \leq b_n$, then the $\sum a_n$ converges to some $\alpha \leq \beta$. (Note: you can also guarantee convergence if $|a_n| \leq b_n$ for $n>N$.)
Geometric series
$\sum_{k=0} ^\infty = 1 + \lambda + .. + \lambda^n + ...$
Harmonic Series
$\sum_{k=1}^\infty = 1 +\frac{1}{2} + \frac{1}{3} + ...$
p-Test
The series $\sum \frac{1}{n^p}$ from $n=1$ to $\infty$ converges if and only if $p>1$. If the area under the curve is finite, then the function converges. If the area is infinite, then the function does not converge.
Rearrangement Theorem

(1) Let $\sum_{n=1}^\infty a_n$ be a absolutely convergent sequence. Then any rearrangement of terms in that series results in a new series that is also absolutely convergent to the same limit.

(2) Let $\sum_{n=1}^\infty a_n$ be a conditionally convergent sequence. Then, for any real number c there is a rearrangement of the series such that the new resulting series will converge to c.

## Terms/Theorems (Sequences/Series of Functions)

Analytic Function
A function is analytic if it has power series expansions about all points. Ex: Sin(x)
Pointwise Convergence
A sequence of functions $f_n$ : [a,b] $\to \mathbb{R}$ converges pointwise to a limit function $f$ : [a,b] $\to \mathbb{R}$ if for each x $\in$ [a,b], $\lim_{n\to \infty} f_n(x)$ = $f(x)$.
Power Series
$\sum_{k=0} ^\infty = a_0+a_1x+a_2x^2...a_nx^n$
Uniform Convergence
A sequence $(f_n)$ converges uniformly to a function $f$ if $\forall \epsilon>0$, $\exists n$ such that for $n>N$, $|f_n(x)-f(x)|<\epsilon$ $\forall x \in$ domain.
Uniform Convergence Theorem
If the sequence $f_n$ converges uniformly to $f$ and $f_n$ is continuous, then $f$ is continuous.