Chapter 2 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.