620-295 Real Analysis with Applications

Assignment 4: Due 5pm on 18 September

Lecturer: Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville VIC 3010 Australia
aram@unimelb.edu.au

Due 5pm on 18 September in the appropriate assignment box on the ground floor of Richard Berry.

1. Define the following and give an example for each:

   (a)   metric space,

   (b)   complete (for a metric space),

   (c)   completion (of a metric space),

   
   
2. Let $X$ be a metric space and let $\left(a_n\right)$ be a sequence in $X$. Show that if $\left(a_n\right)$ converges then $\left(a_n\right)$ is Cauchy.
   
   
3. Let $\left(a_n\right)$ be a sequence in ${ℝ}_{\ge 0}$. Show that if ${\displaystyle \sum _{n=1}^{\infty }}\left|a_n\right|$ converges then ${\displaystyle \sum _{n=1}^{\infty }}{a}_n$ converges.
   
   
4. Prove that $\underset{x\to 0}{\lim }{\displaystyle \frac{\cos x-1}{x}}=0$ and $\underset{x\to 0}{\lim }{\displaystyle \frac{\cos x-1}{x^2}}=-\frac{1}{2}$.
   
   
5. Define the following and give an example for each:

   (a)   metric space,

   (b)   limit of $f$ as $x$ approaches $a$,

   (c)   limit of $\left(x_n\right)$ as $n\to \infty$,

   (j)   continuous at $x=a$,

   (c)   continuous,

   (d)   uniformly continuous,

   (e)   Lipschitz,

   (f)   $ϵ$-ball,

   
   
6. Define the following and give an example for each:

   (a)   topology,

   (b)   topological space,

   (c)   open set,

   (d)   closed set,

   (e)   interior,

   (f)   closure,

   (g)   interior point,

   (h)   close point,

   (i)   neighborhood.

   
   
7. Let $X$ and $Y$ be metric spaces. Define the topology on $X$ and $Y$. Define carefully what it means for $f:X\to Y$ to be continuous as a function between metric spaces and define carefully what it means for $f:X\to Y$ to be continuous as a function between topological spaces.
   
   
8. Let $a,b\in ℝ$ and let $f:[a,b]\to ℝ$ be a function. Let $c\in [a,b]$ and carefully define $f\prime \left(c\right)$. Prove that if $f:[a,b]\to ℝ$ and $g:[a,b]\to ℝ$ are functions then $\left(fg\right)\prime \left(c\right)=f\left(c\right)g\prime \left(c\right)+f\prime \left(c\right)g\left(c\right)$, whenever $f\prime \left(c\right)$ and $g\prime \left(c\right)$ exist.