## 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.