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\noindent
\parbox{2.25in}{\bf Math 3220}
\hfill {\Large \bf Final Revisions} \hfill
\parbox{2.25in}{\bf \hfill Due: Final Exam}

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%\noindent {\bf\large Name:} \underline{\qquad\sc\large This is my fabulous name! \qquad}
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\begin{description}
\item[Homework \#1 Problem \#4] Let $N$ be a positive integer and $x_1,\dots,x_N \in \mathbb{R}$. 

Show that $|x_1+\cdots+x_N| \leq |x_1|+\cdots+|x_N|$.  \hfill  {\it Note:} Prove this using induction on $N$.

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\item[Homework \#2 Problem \#3] Let $S \subseteq \mathbb{R}$ be non-empty and bounded below. Let $-S = \{ -x \;|\; x \in S\}$. Show that $\mathrm{sup}(-S)$ exists. Then show that $-\mathrm{inf}(S) = \mathrm{sup}(-S)$.

This problem shows that the completeness axiom guaranteeing the existence of supremums implies a similar statement about the existence of infimums. Write down an ``infimum'' version of the completeness axiom.

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\item[Homework \#3 Problem \#1] Prove that the following sequences converge. Use the definition of convergence. Don't use any ``fancy'' theorems.
\begin{enumerate}[(a)]
\item $\displaystyle \left\{ \dfrac{5n}{n+1} \right\}_{n=1}^\infty$

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\item $\displaystyle \left\{ \dfrac{2n+1}{n^3+3n-1} \right\}_{n=1}^\infty$

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\item[Homework \#4 Problem \#3] Let $\displaystyle \sum_{k=1}^\infty  a_k$ and $\displaystyle \sum_{k=1}^\infty b_k$ be convergent series.
\begin{enumerate}[(a)]
\item Prove that $\displaystyle \sum_{k=1}^\infty (a_k+b_k)$ converges to $\displaystyle \sum_{k=1}^\infty  a_k + \sum_{k=1}^\infty b_k$.

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\item Suppose that $\displaystyle \sum_{k=1}^\infty  c_k$ diverges. Can we conclude that $\displaystyle \sum_{k=1}^\infty (a_k + c_k)$ diverges as well? 

Prove this or give a counter-example.

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\item[Homework \#5 Problem \#5] Let $A,B \in \mathbb{R}$ and define $f:\mathbb{R} \to \mathbb{R}$ by $f(x)=Ax+B$. 

Prove that $f$ is continuous (everywhere). Do this using the $\epsilon$-$\delta$ definition of continuity.

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\item[Homework \#6 Problem \#4] Let $f(x)$ be differentiable at $x=a$. Let $k$ be some constant. 

Show that $(kf)'(a) = k\,f'(a)$ (i.e., we can pull constants out of derivatives).

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\item[Homework \#7 Problem \#2] If we have a function $f:\mathbb{R} \to \mathbb{R}$ such that $|f(x)-f(y)| \leq |x-y|$ for all $x,y \in \mathbb{R}$, then we say that $f$ is a {\it contraction}.
\begin{enumerate}[(a)]
\item Let $f$ be a contraction. Show that $f$ is uniformly continuous.

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\item Suppose that $f$ is differentiable everywhere and that $|f'(x)| \leq 1$ for all $x \in \mathbb{R}$. Show that $f$ is a contraction.

{\it Suggestion:} MVT.

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\item[Homework \#8 Problem \#4 Linear] Let $f$ and $g$ be integrable on $[a,b]$. 

Use the definition of the Riemann integral to show that $f+g$ is integrable with $\int_a^b (f+g) = \int_a^b f + \int_a ^b g$.

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\end{description}

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