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\title{Math 112 Problem Set 1}
\author{Due Jan. 31, 2018}

\maketitle
 

{\bf Core problems:}
\begin{enumerate}
	\item Show that 
	\begin{enumerate}
		\item $f:S\to T$ is one-to-one iff there is a function $g:T \to S$ such that $g \circ f = \id_S$;
		\item $f: S \to T$ is onto iff there is a function $h:T \to S$ such that $f \circ h = \id_T$; and
		\item $f:S \to T$ is a bijection iff there is a map $g:T\to S$ such that $f\circ g = \id_T$ and $g\circ f = \id_S$.  Show also that $g= f^{-1}$ is uniquely determined.
	\end{enumerate}
	Recall that $I_S$ is the function with domain and range $S$ such that, for all $x \in S$, $I_S(x) = x$. (Intro, \#15)
	\item Prove that, in an ordered field, if $\sqrt{2}$ is a positive number whose square is $2$, then $\sqrt{2} < \frac{3}{2}$. (1.1, \#4)
	\item Define $\{x_n\}_n$ inductively by $x_n=0$ and $x_{n+1} = \sqrt{2+x_n}$ and set $\lambda = \lim_{n\to \infty} x_n$ (Example 1.2.10 shows that this limit exists).
	\begin{enumerate}
		\item Show that $\lambda$ is a root of $\lambda^2 - \lambda - 2 = 0$.
		\item Find $\lim_{n\to \infty} x_n$. (1.2, \#1)
	\end{enumerate}
	\item Show that $\frac{3^n}{n!}$ converges to $0$. (1.2, \#2)

	\item (Moved to next week)
	\item (Moved to next week)
	\item \begin{enumerate}
		\item Let $x \geq 0$ be a real number such that, for any $\epsilon > 0$, $x \leq \epsilon$.  Show that $x = 0$.
		\item Let $S = (0, 1) \subset \R$.  Show that for every $\epsilon > 0$, there is an $x \in S$ such that $x < \epsilon$.
	\end{enumerate} (1, \#3)
	\item (Moved to next week)
	\item Show that, in $\R$, $\lim_{n\to \infty}x_n = x$ iff $\lim_{n \to \infty}(-x_n) = -x$.  Prove that the completeness axiom is equivalent to the statement that every decreasing sequence $x_1 \geq x_2\geq\dots$ bounded below converges.  YOU DO NOT NEED TO PROVE IT CONVERGES TO A PARTICULAR VALUE(1, \#18)
	\item Look at Propositions 1.1.2, 1.1.5, and 1.1.6.  Pick two items that aren't proved in the book and weren't proved in class and prove them.
\end{enumerate}

{\bf Niche problems:}
\begin{enumerate}
	\item \begin{enumerate}
		\item Define $\phi: \N \times \N \to \N$ by $\phi(i, j) = j+ \frac{1}{2}k(k+1)$ where $k = i+j$.  Show that $\phi$ is a bijection and that it has something to do with [the picture in the book].
		\item Show that if $A_1, A_2, \dots$ are countable sets, then so is $\cup_{n=1}^\infty A_n$.
	\end{enumerate} (Intro, \#8)
	\item Let $x_n = \sqrt{n^2+1}-n$.  Compute $\lim_{n\to \infty} x_n$. (1.2, \#3)
		\item For each $x \in \R$ with $x \geq 0$, show that there is a $y \in \R$ such that $y^2 = x$.
		\item Prove that $\Z$ is countable.

\end{enumerate}

{\bf Doc Brown problems:}
\begin{enumerate}
	\item Find a sequence $\{x_n\}_n$ such that $\limsup_n x_n = 5$ and $\liminf_n x_n = 3$. (1.5, \#2)
	\item What is the angle between $(3,2,2)$ and $(0,1,0)$? (1.6, \#2)
	\item Put the inner product $\langle f, g\rangle = \int_0^1 f(x) g(x) dx$ on $\cC([0,1])$.  Verify the Cauchy-Shwarz inequality for $f(x) = 1$ and $g(x) = x$. (1.7, \#3)
\end{enumerate}

Note that I have written out the problems that appear in Marsden and Hoffman because it's still shopping period.  You shouldn't rely on this in future problem sets.
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