handouts/Advanced/Nonstandard Analysis/parts/supremum.tex

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\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