Homework 12, due Wednesday 11 April. Problem 5.1.

Determine which of the following functions $f:\RR\to\RR$ are differentiable at $0$.
\begin{enumerate}[(i)]
\item $f(x)=x\sin(\frac{1}{x})$ for $x\ne 0$ and $f(0)=0$.
\item $f(x)=x^{\frac{4}{3}}\sin(\frac{1}{x})$ for $x\ne 0$ and $f(0)=0$.
\item $f(x)=x^2$ for $x\le 0$ and $f(x)=x^3$ for $x>0$.
\item $f(x)=x^2$ for $x\le 0$ and $f(x)=x$ for $x>0$.
\end{enumerate}

Homework 11, due Friday 6 April. Problem 4.7.

Suppose $f:[0,\infty)\to \mathbb R$ is continuous. Show, if there is a $b>0$ such that $f|_{[b,\infty)}$ is uniformly continuous, then $f$ is uniformly continuous.

Show that $f:[0,\infty)\to \mathbb R$ defined by $f(x)=\sqrt{x}$ is uniformly continuous. (Compare with Example 4.44.)

Homework 10, due Monday 2 April. Problem 4.17.

Suppose $A,B, D\subset \mathbb R$ and $f:D\to \mathbb R$. Show
\begin{enumerate}[(i)]
\item $f^{-1}(\tilde{A}) = \widetilde{f^{-1}(A)}$;
\item $f^{-1}(A\cup B) = f^{-1}(A) \cup f^{-1}(B)$;
\item $f^{-1}(A\cap B)= f^{-1}(A)\cap f^{-1}(B)$.
\end{enumerate}

Homework 9, due Wednesday 28 March. Problem 4.15.

Homework 8, due Monday 12 March. Problem 3.15.

\def\RR{\mathbb{R}}
Define $f:\RR\setminus \{0\}\to\RR$ by $f(x) = x\sin(\frac{1}{x})$ and define $g:\RR\to\RR$ by
\[
g(x) = \begin{cases} & 1 \mbox{ if } x=0 \\
& 0 \mbox{ if } x\ne 0.
\end{cases}
\]
Show
\[
\lim_{x\to 0} f(x) = 0 =\lim_{x\to 0} g(x),
\]
but $g\circ f:\RR\setminus\{0\} \to\RR$ does not have a limit at $0$. Discuss carefully the relation of this example to Proposition 3.33.

Homework 7, due Monday 26 February. Problems 3.1(b).

Define $f:\mathbb{R}\to \mathbb{R}$ by $f(x)=x^3$. Show $f$ has limit $1$ at $1$.

Homework 6, due Wednesday 21 February. Problem 2.12.

Let $F_0=0$ and $F_1=1$ and define, recursively,
\[
F_{n+1} = F_n+F_{n-1}
\]
(the Fibonacci sequence). Let $a_n = \frac{F_{n+1}}{F_n}$. Is the sequence $(a_n)$ monotone?

Show $a_{n+1}a_n \ge 2$. Show,
\[
|a_{n+1}-a_n| = \frac{|a_{n-1}-a_n|}{a_n a_{n-1}}.
\]
Conclude that $(a_n)$ converges. Identify the limit.

Homework 5, due Friday 16 February.

Let a_1=1 and define, for positive integers n, a_{n+1} = sqrt(1+a_n). Show (a_n) converges to the golden ratio.

Latex code below.

Define a real sequence recursively as follows. Let $a_1= 1$ and $a_{n+1} = \sqrt{1+a_n}$ for $n\in \mathbb N^+.$ Show that $(a_n)$ converges to the golden ratio $\frac{\sqrt{5}+1}{2}$.

Homework 4, due Wednesday 31 January.

Let (a_n)_{n=3}^\infty denote the sequence a_n = (2n^2+5)/(n^2-3n+2). Prove, directly from the definition of limit, that (a_n) converges to 2.

Latex code below.

Let $(a_n)_{n=3}^\infty$ denote the sequence
\[
a_n = \frac{2n^2+5}{n^2-3n+2}.
\]
Prove, directly from the definition of limit, that $(a_n)$ converges to $2$.

Homework 3, due Friday 26 January. Determine, with proof, the accumulation points of the set (0,1].

Homework 2, due Friday 26 January. Problem 1.5.

Let $S=\{\frac{1}{n}:n\in\mathbb N^+\}. Show $S$ has a greatest lower bound and $\glb(S)=0.$

Homework 1, due Friday 19 January. Problem 1.1 (Does not include showing that the integers mod 3 is a field. Rather take that fact as given.)

Let $\mathbb Z_3 =(\{0,1,2\},+,\cdot)$ where
\[
\begin{split}
x+y=& x+y \mbox{ modulo } 3 \\
xy=& xy \mbox{ modulo } 3.
\end{split}
\]
It is easy, but tedious, to verify that $\mathbb Z_3$ is a field.

Find the additive inverse for $1$ and multiplicative inverse for $2$.

Show there is no order $<$ on $\{0,1,2\}$ such that $(\mathbb Z_3,<)$
is an ordered field. (Suggestion: Arguing by contradiction, show the additive
inverse for $1$ would have to be both positive and negative.)