# Additional Terms

## 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.

## Guidelines:

- Each student is required to add a minimum of two entries to each glossary.
- Entries should be
*in your own words*and not copy/pasted from elsewhere, or typed verbatim from the text. - Proper mathematical formatting using TeX is required.
- Include any terms which
*"move the story along"*, or the ones that seem to show up again and again.

## How to edit this page

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**Example**- Use colons to define the term. Include spaces between the colons and the surrounding words. Use double stars around the title of the term to make it
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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 $\lambda$v 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$\to$B is a surjection if for each b$\in$B 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
that are both closed and open. Fact: In $\mathbb{R}$ the only clopen sets are $\varnothing$ and $\mathbb{R}$ .*M* **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
, the elements of which are referred to as points of*M*, together with the metric*M***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
is connected if it is not disconnected—-that is it contains no proper clopen subset.*M* **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 $\sqcup$ A
^{C} **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:

- The $\epsilon, \delta$ definition
- The inverse image of any closed set in N is closed in M.
- 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.