153 lines
4.0 KiB
TeX
153 lines
4.0 KiB
TeX
% Copyright (C) 2023 <Mark (mark@betalupi.com)>
|
|
%
|
|
% This program is free software: you can redistribute it and/or modify
|
|
% it under the terms of the GNU General Public License as published by
|
|
% the Free Software Foundation, either version 3 of the License, or
|
|
% (at your option) any later version.
|
|
%
|
|
% You may have received a copy of the GNU General Public License
|
|
% along with this program. If not, see <https://www.gnu.org/licenses/>.
|
|
%
|
|
%
|
|
%
|
|
% If you edit this, please give credit!
|
|
% Quality handouts take time to make.
|
|
|
|
|
|
\section*{The supremum \& the infimum}
|
|
|
|
\definition{}
|
|
In this section, we'll define a \say{real number} as a decimal, infinite or finite.
|
|
|
|
\problem{}
|
|
Write $2.317171717...$ as a simple fraction.
|
|
|
|
\vfill
|
|
|
|
\problem{}
|
|
Write $\nicefrac{2}{11}$ as an infinite decimal and prove that your answer is correct.
|
|
|
|
\vfill
|
|
|
|
\problem{}
|
|
Show that $0.999... = 1$
|
|
|
|
\note[Note]{
|
|
There is no real number $0.0...1$ with a digit $1$ \say{at infinity.} \\
|
|
Some numbers have two decimal representations, some have only one.
|
|
}
|
|
|
|
|
|
\vfill
|
|
|
|
|
|
\problem{}
|
|
Concatenate all the natural numbers in order to form $0.12345678910111213...$. \par
|
|
Show that the resulting decimal is irrational.
|
|
|
|
\vfill
|
|
|
|
\problem{}
|
|
Show that a rational number exists between any two real numbers.
|
|
|
|
\vfill
|
|
\pagebreak
|
|
|
|
\definition{}
|
|
Let $M$ be a subset of $\mathbb{R}$.\par
|
|
We say $c \in \mathbb{R}$ is an \textit{upper bound} of $M$ if $c \geq m$ for all $m \in M$. \par
|
|
The smallest such $c$ is called the \textit{supremum} of $M$, and is denoted $\text{sup}(M)$. \par
|
|
|
|
\vspace{2mm}
|
|
|
|
Similarly, $x \in \mathbb{R}$ is a \textit{lower bound} of $M$ if $x \leq m$ for all $m \in M$. \par
|
|
The largest upper bound of $M$ is called the \textit{infimum} of $M$, denoted $\text{inf}(M)$.
|
|
|
|
\problem{}
|
|
Show that $x$ is the supremum of $M$ if and only if...
|
|
\begin{itemize}
|
|
\item for all $m \in M$, $m \leq x$, and
|
|
\item for any $x_0 < x$, there exists an $m \in M$ so that $m > x_0$
|
|
\end{itemize}
|
|
|
|
\vfill
|
|
|
|
|
|
|
|
\problem{}
|
|
Show that any subset of $\mathbb{R}$ has at most one supremum and at most one infimum.
|
|
|
|
\vfill
|
|
|
|
|
|
|
|
\problem{}
|
|
Find the supremum and infimum of the following sets:
|
|
\begin{itemize}
|
|
\item $\bigl\{ a^2 + 2a \bigl| -5 < a < 5\bigr\}$
|
|
\item $\bigl\{\pm \frac{n}{2n + 1} \bigl| n \in \mathbb{N}\bigr\}$
|
|
\end{itemize}
|
|
|
|
\vfill
|
|
|
|
|
|
\problem{}
|
|
Let $A$ and $B$ be subsets of $\mathbb{R}$, and let $\text{sup}(A)$ and $\text{sup}(B)$ be known. \par
|
|
Compute the following in terms of $\text{sup}(A)$ and $\text{sup}(B)$.
|
|
\begin{itemize}
|
|
\item $\text{sup}(A \cup B)$
|
|
\item $\text{sup}(A + B)$, where $A + B = {a + b \forall (a, b) \in A \times B}$,
|
|
\item $\text{inf}(A \cdot B)$, where $A \cdot B = {ab \forall (a, b) \in A \times B}$
|
|
\end{itemize}
|
|
|
|
|
|
\vfill
|
|
|
|
\problem{}
|
|
Prove the assumptions in \ref{stpart}: \par
|
|
Show that $\text{st}(x)$ is exists and is unique for limited $x$.
|
|
|
|
\vfill
|
|
\pagebreak
|
|
|
|
|
|
\theorem{Completeness Axiom}<completeness>
|
|
Every non-empty subset of $\mathbb{R}$ that is bounded above has a least upper bound.
|
|
|
|
\problem{}
|
|
Show that $a < \text{sup}(A)$ if and only if there is a $c$ in $A$ where $a < c$
|
|
|
|
\vfill
|
|
|
|
\problem{}
|
|
Use the definitions in this handout to prove \ref{completeness}. \par
|
|
\hint{Build the supremum one digit at a time.}
|
|
|
|
\vfill
|
|
|
|
\problem{}
|
|
Let $[a_1, b_1] \supseteq [a_2, b_3] \supseteq [a_3, b_3] \supseteq ...$ be an infinite sequence of closed line intervals.
|
|
\par Show that there exists a $c \in \mathbb{R}$ that lies in all of them. Is this true of open intervals?
|
|
|
|
|
|
\vfill
|
|
\pagebreak
|
|
|
|
\problem{Bonus}
|
|
Show that every real number in $[0, 1]$ can be written as a sum of 9 numbers \par
|
|
Whose decimal representations only contain 0 and 8. \par
|
|
|
|
\vfill
|
|
|
|
\problem{Bonus}
|
|
Two genies take an infinite amount of turns and write the digits of an infinite
|
|
decimal. The first genie, on every turn, writes any finite amount of digits to the tail of the decimal.
|
|
The second genie writes one digit to the end. If the resulting decimal after an infinite amount
|
|
of turns is periodic, the first genie wins; otherwise, the second genie wins. Who has a winning
|
|
strategy? \par
|
|
|
|
|
|
\vfill
|
|
\pagebreak
|
|
|