Real Analysis

Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia
aram@unimelb.edu.au

Last update: 20 July 2014

Lecture 32

Let $X$ and $Y$ be topological spaces.
Let $f:X\to Y$ be a continuous function.
Let $E\subseteq X\text{.}$

(a)	If $E$ is connected then $f\left(E\right)$ is connected.
(b)	If $E$ is compact then $f\left(E\right)$ is compact.

Let $X={ℝ}^1\text{.}$ Let $E\subseteq X\text{.}$

(a)	$E$ is connected if and only if $E$ is an interval.
(b)	$E$ is compact and connected if and only if there exist $m,M\in ℝ$ such that $E=[m,M]\text{.}$

(Intermediate value theorem) Let $f:[a,b]\to ℝ$ be continuous. Then there exist $m,M\in ℝ$ such that $$f\left([a,b]\right)=[m,M]\text{.}$$

(Rolle's theorem) Let $f:[a,b]\to ℝ$ be continuous with $f\left(a\right)=f\left(b\right)\text{.}$ If $f\prime :(a,b)\to ℝ$ exists then there exists $c\in (a,b)$ such that $f\prime \left(c\right)=0\text{.}$

Idea that makes it work: If $M$ is the maximum value of $f$ and $f\left(c\right)=M$ then $f\prime \left(c\right)=0\text{.}$

(The mean value theorem) Let $f:[a,b]\to ℝ$ be a continuous function such that $f\prime :(a,b)\to ℝ$ exists. Then there exists $c\in (a,b)$ such that $$f\left(b\right)=f\left(a\right)+f\prime \left(c\right)(b-a)\text{.}$$

(Taylor's theorem) Let $f:[a,b]\to ℝ$ be a function such that $f^{\left(N\right)}:(a,b)\to ℝ$ exists. Then there exists $c\in (a,b)$ such that $$\begin{array}{ccc}{\displaystyle f\left(b\right)}& =& {\displaystyle f\left(a\right)+f\prime \left(a\right)(b-a)+\frac{1}{2!}f″\left(a\right){(b-a)}^2}\\ & & {\displaystyle +\frac{1}{3!}{f}^{\left(3\right)}\left(a\right){(b-a)}^2+\cdots +\frac{1}{(N-1)!}{f}^{(N-1)}\left(a\right){(b-a)}^{N-1}}\\ & & {\displaystyle +\frac{1}{N!}{f}^{\left(N\right)}\left(c\right){(b-a)}^N\text{.}}\end{array}$$

If we let $a=0$ and $b=x$ then Taylor's theorem says that $f\left(x\right)$ can be expanded as a polynomial in $x,$ $$f\left(x\right)=f\left(0\right)+f\prime \left(0\right)x+\frac{1}{2!}f″\left(0\right)x^2+\frac{1}{3!}{f}^{\left(3\right)}\left(0\right)x^3+\frac{1}{4!}{f}^{\left(4\right)}\left(0\right)x^4+\cdots$$ You need more and more terms for more and more accuracy in the expansion. Just like... You need more and more digits $$\pi =3.141592\dots$$ for more and more accuracy in $ℝ\text{.}$

When does an infinite polynomial like $$\frac{1}{1-x}=1+x+x^2+x^3+x^4+\cdots$$ produce a good function? i.e. For which $a\in ℂ$ does the series $\sum _{n=0}^{\infty }{x}^n$ converge. In other words: Let $s_1=1,$ $s_2=1+a,$ $s_3=1+a+a^2,$ $s_4=1+a+a^2+a^3,\dots$ For which $a\in ℂ$ does the sequence $(s_1,s_2,s_3,\dots )$ converge? In other words: Does $\underset{n\to \infty }{\text{lim}}{s}_n$ exist? In other words: Does there exist $\ell \in ℂ$ such that
if $\epsilon \in {ℝ}_{>0}$ then there exists $N\in {\mathbb{Z}}_{>0}$ such that
if $n\in {\mathbb{Z}}_{>0}$ and $n>N$ then $d(s_n,\ell )<\epsilon \text{.}$ In this example: $$\begin{array}{ccc}{\displaystyle s_1}& =& {\displaystyle 1,}\\ {\displaystyle s_2}& =& {\displaystyle 1+a,}\\ {\displaystyle s_3}& =& {\displaystyle 1+a+a^2=\frac{1-a^3}{1-a},}\\ {\displaystyle s_4}& =& {\displaystyle 1+a+a^2+a^3=\frac{1-a^4}{1-a},}\\ {\displaystyle }& ⋮& {\displaystyle }\\ {\displaystyle s_n}& =& {\displaystyle 1+a+a^2+\cdots +a^{n-1}=\frac{1-a^n}{1-a}\text{.}}\end{array}$$ So $$\begin{array}{ccc}{\displaystyle \underset{n\to \infty }{\text{lim}}{s}_n}& =& {\displaystyle \underset{n\to \infty }{\text{lim}}\frac{1-a^n}{1-a}}\\ & =& {\displaystyle \frac{1}{1-a}\underset{n\to \infty }{\text{lim}}1-a^n}\\ & =& {\displaystyle \frac{1}{1-a}(1-\underset{n\to \infty }{\text{lim}}{a}^n),}\end{array}$$ where we have used the limit theorems:

Let $\left(a_n\right)$ and $\left(b_n\right)$ be sequences (in $ℝ$ or in ${ℝ}^n\text{)}$ and assume $\underset{n\to \infty }{\text{lim}}{a}_n$ and $\underset{n\to \infty }{\text{lim}}{b}_n$ exist. Then

(a)	$\underset{n\to \infty }{\text{lim}}(a_n+b_n)=\underset{n\to \infty }{\text{lim}}{a}_n+\underset{n\to \infty }{\text{lim}}{b}_n,$
(b)	if $c\in ℝ$ then $\underset{n\to \infty }{\text{lim}}c{a}_n=c\underset{n\to \infty }{\text{lim}}{a}_n,$
(c)	$\underset{n\to \infty }{\text{lim}}\left(a_n{b}_n\right)=\left(\underset{n\to \infty }{\text{lim}}{a}_n\right)\left(\underset{n\to \infty }{\text{lim}}{b}_n\right),$
(d)	if $b_n\ne 0$ for all $n\in {\mathbb{Z}}_{>0}$ then $$\underset{n\to \infty }{\text{lim}}\left(\frac{a_n}{b_n}\right)=\frac{\left(\underset{n\to \infty }{\text{lim}}{a}_n\right)}{\left(\underset{n\to \infty }{\text{lim}}{b}_n\right)}\text{.}$$

Skills:

(1)	Algebraic manipulations - Expressions
(2)	Graphing
(3)	Limits
(4)	Sequences (functions ${\mathbb{Z}}_{>0}\to ℝ$ and ${\mathbb{Z}}_{>0}\to ℂ\text{)}$
(5)	Series (a special kind of sequence)
(6)	Approximations - Taylor's theorem and Integral approximations
(7)	Mathematical Language and Proof machine: Fields, Ordered fields, Open sets, Inductions, Definitions and Proof machine

Notes and References

These are notes from a 2010 course on Real Analysis 620-295. This page comes from 100524Lect32.pdf and was given on 24 May 2010.

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