sequence, series, function

Prerequisites

Definition

If there is no additional explanation and the sequence is not in a field, it is assumed that the sequence is in the metric space.
If there is an operation on a field, it is assumed that the sequence is in $\mathbb{C}$.

$\forall x\in X,\forall r>0,B(x,r):=\{y\in X:d(x,y)<r\}$ is called the open neighborhood ball(o.n.b.) of $x$.

$E\subseteq X$ is bounded if there exists $x\in X$ and $r>0$ such that $E\subseteq B(x,r)$.

Let $E$ be a nonempty subset of a metric space $X$, and let $S$ be the set of all real numbers of the form $d(p,q)$, with $p\in E$ and $q\in E$. The sup of $S$ is called the diameter of $E$.

Properties

1. $\{p_n\}$ be a sequence in a metric space $X$

• $\{p_n\}$ converges to $p\in X$ if and only if every o.n.b. of $p$ contains $p_n$ for all but finitely many $n$.
• If $p\in X$, $p^{\prime }\in X$, and if $\{p_n\}$ converges to $p$ and to $p^{\prime }$, then $p^{\prime }=p$.
• If $\{p_n\}$ converges, then $\{p_n\}$ is bounded.
• If $E\subseteq X$ and if $p$ is a limit point of $E$, then there is a sequence $\{p_n\}$ in $E$ such that $p={\lim }_{n\to \infty }{p}_n$.

2. $\{a_n\}$ and $\{b_n\}$ are complex sequences, if $a_n,b_n$ converges to $x,y$ respectively, then the following statements are true:

• $a_n+b_n$ converges to $x+y$
• $a_n{b}_n$ converges to $xy$
• if $y\ne 0\wedge b_n\ne 0$, then $\frac{1}{b_n}$ converges to $\frac{1}{y}$
• if $y\ne 0\wedge b_n\ne 0$, then $\frac{a_n}{b_n}$ converges to $\frac{x}{y}$
• $a_n\to 0$ $⟺$ $|a_n|\to 0$
• $a_n+i{b}_n\to x+iy$ $⟺$ $a_n\to x\wedge b_n\to y$

3. Every bounded sequence in ${\mathbb{R}}^k$ contains a convergent subsequence.
4. The subsequential limits of a sequence $\{p_n\}$ in a metric space $X$ form a closed subset of $X$.
5. If $E$ is the closure of a set $E$ in a metric space $X$, then $\text{diam}\overline{E}=\text{diam}E$.
6. If $\{K_n\}$ is a sequence of compact sets in $X$ such that $K_n\supseteq K_{n+1}$ for all $n$, and if ${\lim }_{n\to \infty }\text{diam}{K}_n=0$, then ${\bigcap }_{n=1}^{\infty }{K}_n$ consists of exactly one point.
7. Let $\{a_i\}$ be a complex sequence, $\sum a_n$ converges if and only if for every $ϵ>0$ there is an integer $N$ such that $\forall m,n,N<m\le n$, ${\sum }_{n=m}^n|a_n|<ϵ$. lf $\sum a_n$ converges, then ${\lim }_{n\to \infty }{a}_n$ = 0.
8. Let $f$ be monotonically increasing on $(a,b)$. Then $f(x+)$ and $f(x-)$ exist at every point of $x$ of $(a,b)$. More precisely, ${\sup }_{a<t<x}f(t)=f(x-)\le f(x)\le f(x+)={\inf }_{x<t<b}f(t)$. Furthermore, if $a<x<y<b$, then $f(x+)\le f(y-)$.
9. Let $f$ be monotonic on $(a,b)$. Then the set of points of $(a,b)$ at which $f$ is discontinuous is at most countable.