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{\bf \Large Math 112 Homework 8}\medskip

Due Thursday April 18, 2019, on Canvas.
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You are encouraged to discuss the homework problems with 
other students. However, what you hand in should reflect your 
own understanding of the material. You are NOT allowed to copy solutions 
from other students or other sources. {\it No late homeworks will be 
accepted.}

Please pay attention to the clarity and precision of your answers. Your
solutions to the problems should always consist of carefully written 
mathematical arguments. 

\medskip

{\bf Material covered:} Rudin pages 120--135, 143--150 (through Theorem 7.12).
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{\bf Problem 1.} (3 points) Rudin Chapter 6 Problem 4.
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{\bf Problem 2.} (5 points) Rudin Chapter 6 Problem 8.

(Hint: Choose a partition of $[1,n]$ so that $U(P,f)$ and $L(P,f)$ are
related to partial sums of the series).
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{\bf Problem 3.} (6 points) Rudin Chapter 6 Problem 10 parts (a)(b)(c),
with all Stieltjes integrals ($\int \dots d\alpha$) replaced by Riemann
integrals ($\int \dots dx$). 

{\it Hint:} For part (a), you can e.g.\ study the
behavior of the function defined by $\phi(u)=\frac{1}{p}u^p-uv$ for a 
fixed value of $v$.
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{\bf Problem 4.} (5 points) 

The following result is a slightly different version of Taylor's theorem,
where the error term is expressed as an explicit integral:

Let $I\subset\mathbb{R}$ be an interval, and let $f:I\to\mathbb{R}$ be a 
function differentiable $(n+1)$ times, with all its derivatives continuous.
Given $a,\,x\in I$, let $$R_{n+1}(x)=\frac{1}{n!}\int_a^x (x-t)^n
f^{(n+1)}(t)\,dt.$$ Then
$$f(x)=\sum_{k=0}^n \frac{f^{(k)}(a)}{k!}(x-a)^k+R_{n+1}(x).$$

(a) From which theorem does the case $n=0$ follow?

(b) Assume by induction that the result is true for $n-1$. Use
integration by parts on $R_n(x)$ to prove the theorem.
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{\bf Problem 5.} (5 points)

If $\{f_n\}$ and $\{g_n\}$ are sequences of bounded functions which converge
uniformly on some set $E$, prove that $\{f_n g_n\}$ converges uniformly on
$E$. Show that the result becomes false if one does not require the
functions $f_n$ and $g_n$ to be bounded.
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{\bf Problem 6.} (5 points) Rudin Chapter 7 Problem 4. (Assume
$x\ge 0$ throughout the problem).%

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