620-295 Real Analysis with applications

Problem Sheet 4

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

Last updates: 9 September 2009

Sequences and series

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 $\left(a_n\right)$ be a sequence in $ℝ$. Show that if $\left(a_n\right)$ is increasing and bounded then $\left(a_n\right)$ converges.
   
   
3. Let $\left(a_n\right)$ be a sequence in ${ℝ}_{\ge 0}$. Show that if ${\displaystyle \sum _{n=1}^{\infty }{a}_n}$ is bounded then ${\displaystyle \sum _{n=1}^{\infty }{a}_n}$ converges.
   
   
4. 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.
   
   
5. Give an example of a Cauchy sequence that does not converge.
   
   
6. Let $\left(a_n\right)$ be a sequence in ${ℝ}_{\ge 0}$. Show that if ${\displaystyle \sum _{n=1}^{\infty }{a}_n}$ converges then $\underset{n\to \infty }{\lim }\left|a_n\right|=0$.
   
   
7. 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.
   
   
8. Let $r\in ℝ$ with $0<r<1$. Prove that $\underset{n\to \infty }{\lim }{\displaystyle \frac{\log (1+\frac{r}{n})}{\frac{r}{n}}}=1$.
   
   
9. Prove that $\underset{x\to 0}{\lim }{\displaystyle \frac{\log (1+x)}{x}}=1$.
   
   
10. Prove that $\underset{x\to 0}{\lim }{\displaystyle \frac{\sin x}{x}}=1$.
    
    
11. Prove that $\underset{x\to 0}{\lim }{\displaystyle \frac{\cos x-1}{x}}=0$.
    
    
12. Prove that $\underset{x\to 0}{\lim }{\displaystyle \frac{\cos x-1}{x^2}}=-\frac{1}{2}$.
    
    

Limits

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

   (a)   continuous at $p$,

   (b)   $\underset{x\to a}{\lim }f\left(x\right)$,

   (c)   continuous,

   (d)   uniformly continuous,

   (d)   Lipschiz continuous,

   (e)   derivative at $p$,

   
   
2. For each of the following, guess the limit and then prove the guess by using the definition of limit:

   (a)   $\underset{x\to 4}{\lim }(\frac{1}{2}x-3)$,

   (b)   $\underset{x\to 0}{\lim }\frac{1}{1+x}$,

   (c)   $\underset{x\to 4}{\lim }\frac{1}{1+x^2}$,

   (d)   $\underset{x\to 1}{\lim }\frac{x^2-1}{x-1}$,

   (e)   $\underset{x\to 9}{\lim }\frac{x+1}{x^2+1}$,

   (f)   $\underset{x\to \infty }{\lim }\frac{\sin x}{x}$,

   (g)   $\underset{x\to 2}{\lim }\frac{2x^2+3x-8}{x^3-2x^2+x-12}$,

   (h)   $\underset{x\to \infty }{\lim }\frac{\log x+2x}{3x-5}$,

   
   
3. Evaluate the following limits:

   (a)   $\underset{x\to 0}{\lim }x\cos \frac{1}{x^2}$,

   (b)   $\underset{x\to 0}{\lim }(\sqrt{5+x^2}-\sqrt{x^{-2}-1})$,

   (c)   $\underset{x\to 0}{\lim }\frac{\sqrt{1+x}-1}{x}$,

   (d)   $\underset{x\to \infty }{\lim }\frac{x^4+x}{x^4+1}$,

   (e)   $\underset{x\to \infty }{\lim }\frac{7x-1}{x^2}$,

   (f)   $\underset{x\to 0^{+}}{\lim }\frac{\sqrt{x}}{\sqrt{7+\sqrt{x+5}}}$,

   (g)   $\underset{x\to 1}{\lim }\frac{|x-1|+1}{x+|x+1|}$,

   (h)   $\underset{x\to \infty }{\lim }\frac{3x^2+1}{2x+1}$,

   
   
4. Evaluate the following limits:

   (a)   $\underset{x\to 0}{\lim }{\displaystyle \frac{1-\cos x}{x+x^2}}$,
   

   (b)   $\underset{x\to \infty }{\lim }\frac{\log x}{x}$,
   

   (c)   $\underset{x\to 0+}{\lim }\sqrt{x}\hspace{0.17em}\log x$,
   

   (d)   $\underset{x\to 0+}{\lim }\frac{\sqrt{x}}{\log x}$,
   

   (e)   $\underset{x\to 0}{\lim }\frac{\sin x}{x}$,
   

   (f)   $\underset{x\to 0}{\lim }(\frac{1}{\arcsin x}-\frac{1}{\sin x})$.

   
   

Continuous functions

1. Let $f:ℝ\to ℝ$ be such that $f$ is continuous at $x=0$ and if $x,y\in ℝ$ then $f(x+y)=f\left(x\right)f\left(y\right)$. Show that if $a\in ℝ$ then $f$ is continuous at $x=a$.
   
   
2. Let $f:{ℝ}_{>0}\to ℝ$ be such that $f$ is continuous at $x=1$ and if $x,y\in {ℝ}_{>0}$ then $f\left(xy\right)=f\left(x\right)+f\left(y\right)$. Show that if $a\in {ℝ}_{>0}$ then $f$ is continuous at $x=a$.
   
   
3. Let $I$ be an interval in $ℝ$. Let $f:I\to ℝ$ be continous. Show that the function $\left|f\right|:I\to ℝ$ given by $\left|f\right|\left(x\right)=\left|f\right(x\left)\right|$ is continuous.
   
   
4. Let $I$ be an interval in $ℝ$ and let $f:I\to ℝ$ and $g:I\to ℝ$ be continous. Show that the function $\max (f,g):I\to ℝ$ given by $\max (f,g)\left(x\right)=\max \left(f\right(x),g(x\left)\right)$ is continuous.
   
   
5. (Thomae's function) Let $f:[0,1]\to ℝ$ be given by $f\left(x\right)=\{\begin{array}{cc}\frac{1}{n},& \text{if}\frac{m}{n}\in \mathbb{Q}\text{is reduced,}\hfill \\ 0,& \text{if}x\notin \mathbb{Q}.\hfill \end{array}$
   Show that

   (a)   If $a\notin \mathbb{Q}$ then $f$ is continuous at $x=a$, and

   (b)   If $a\in \mathbb{Q}$ then $f$ is not continuous at $x=a$.

   
   
6. Let $I$ be an interval in $ℝ$ and let $f:I\to ℝ$ and $g:I\to ℝ$ be continous. Show that the function $\min (f,g):I\to ℝ$ given by $\min (f,g)\left(x\right)=\min \left(f\right(x),g(x\left)\right)$ is continuous.
   
   
7. Let $a\in ℝ$ and let $f:ℝ\to ℝ$ be given by $f\left(x\right)=\{\begin{array}{cc}ax,& \text{if}x\le 0,\hfill \\ \sqrt{x},& \text{if}x>0.\hfill \end{array}$
   Show that $f$ is continuous.
   
   
8. Is the function $f:ℝ\to ℝ$ given by $f\left(x\right)=x$ uniformly continuous?
   
   
9. Is the function $f:(0,1)\to ℝ$ given by $f\left(x\right)=\frac{1}{x}$ uniformly continuous?
   
   
10. Is the function $f:({10}^{-4},1)\to ℝ$ given by $f\left(x\right)=\frac{1}{x}$ uniformly continuous?
    
    
11. Is the function $f:(0,1)\to ℝ$ given by $f\left(x\right)=x^2$ uniformly continuous?
    
    
12. Is the function $f:[-1,1]\to ℝ$ given by $f\left(x\right)=\sqrt{1-x^2}$ uniformly continuous?
    
    
13. Is the function $f:(1,\infty )\to ℝ$ given by $f\left(x\right)=\log x$ uniformly continuous?
    
    
14. Is the function $f:(0,\infty )\to ℝ$ given by $f\left(x\right)=\log x$ uniformly continuous?
    
    
15. Let $f:ℝ\to ℝ$ be the function given by $f\left(x\right)=\frac{x}{(1+|x\left|\right)}$. Show that

    (a)   $f$ is continuous,

    (b)   $f$ is uniformly continuous,

    (c)   $\sup \left(f\right(ℝ\left)\right)=1$,

    (d)   There does not exist $x\in ℝ$ such that $f\left(x\right))=1$,

    (e)   $\inf \left(f\right(ℝ\left)\right)=-1$,

    (d)   There does not exist $y\in ℝ$ such that $f\left(y\right))=-1$.

    
    
    
16. Show that the function $f:ℝ\to ℝ$ given by $f\left(x\right)=x^3-6x+3$ has exactly 3 roots.
    
    
17. Let $I$ be an interval in $ℝ$ and let $f:I\to ℝ$ be a continuous function. Prove that $f\left(I\right)$ is an interval.
    
    
18. Let $I$ and $J$ be intervals in $ℝ$ and let $f:I\to J$ be a surjective strictly monotonic continuous function. Prove that the inverse function $g:J\to I$ exists and is strictly monotonic and continuous.
    
    

Differentiability

1. 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.
   
   
2. Let $f:{ℝ}_{>0}\to ℝ$ be such that $f$ is differentiable at $x=1$ and if $x,y\in {ℝ}_{>0}$ then $f\left(xy\right)=f\left(x\right)+f\left(y\right)$. Show that

   (a)   if $c\in {ℝ}_{>0}$ then $f$ is differentiable at $x=c$,

   (b)   if $c\in {ℝ}_{>0}$ then $f\prime \left(c\right)=f\prime \left(1\right)/c$,

   (c)   Show that $f$ is infinitely differentiable.

   
   
3. Let $f:ℝ\to ℝ$ be such that $f$ is differentiable at $x=0$ and if $x,y\in ℝ$ then $f(x+y)=f\left(x\right)f\left(y\right)$. Show that

   (a)   if $c\in ℝ$ then $f$ is differentiable at $x=c$,

   (b)   if $c\in {ℝ}_{>0}$ then $f\prime \left(c\right)=f\prime \left(0\right)f\left(c\right)$,

   (c)   Show that $f$ is infinitely differentiable.

   
   
4. Let $f:ℝ\to ℝ$ be given by $f\left(x\right)=\{\begin{array}{cc}-x^2,& \text{if}x\le 0,\hfill \\ x,& \text{if}x>0.\hfill \end{array}$
   Is $f$ continuous at $x=0$? Is $f$ differentiable at $x=0$?
   
   
5. Let $f:ℝ\to ℝ$ be given by $f\left(x\right)=\{\begin{array}{cc}-x^2,& \text{if}x\le 0,\hfill \\ x^3,& \text{if}x>0.\hfill \end{array}$
   Is $f$ continuous at $x=0$? Is $f$ differentiable at $x=0$?
   
   
6. Let $f:ℝ\to ℝ$ be given by $f\left(x\right)=\{\begin{array}{cc}\frac{\sin x}{x},& \text{if}x<0,\hfill \\ 1+x^2,& \text{if}x\ge 0.\hfill \end{array}$
   Is $f$ continuous at $x=0$? Is $f$ differentiable at $x=0$?
   
   
7. Let $a,b\in ℝ$ and assume that $f:[a,b)\to ℝ$ is differentiable on $(a,b)$ and continuous on $[a,b)$. Assume that the limit $\underset{x\to a+}{\lim }f\prime \left(x\right)=L$ exists. Prove that the right derivative $f_{+}\prime \left(a\right)$ exists and that $f_{+}\prime \left(a\right)=L$.
   
   
8. Let $a,b\in ℝ$ and assume that $f:(a,b)\to ℝ$ is differentiable at $c$. Show that $\underset{h\to 0+}{\lim }{\displaystyle \frac{f(c+h)-f(c-h)}{2h}}$ exists and equals $f\prime \left(c\right)$. Is the converse true?
   
   
9. Prove that $\frac{d}{dx}\sqrt{x}={\displaystyle \frac{1}{2\sqrt{x}}}$.
   
   
10. Prove that $\frac{d}{dx}\arcsin x={\displaystyle \frac{1}{\sqrt{1-x^2}}}$.
    
    

Mean value theorem

1. Use the mean value theorem to prove the following inequalities:

   (a)   $|\sin x-\sin y|\le |x-y|$ for all $x,y\in ℝ$.

   (b)   $|\log x-\log y|\le \frac{1}{2}|x-y|$ for all $x,y\in [2,\infty )$,

   (c)   $|{(x+1)}^{\mathrm{1/5}}-x^{\mathrm{1/5}}|\le {\left(5x^{\mathrm{4/5}}\right)}^{-1}$ for all $x\in {ℝ}_{>0}$.

   
   
   
2. Use the mean value theorem to show that if a function $f:(a,b)\to ℝ$ is differentiable with $f\prime \left(x\right)>0$ for all $x$ then $f$ is strictly increasing.
   
   
3. Use the mean value theorem to show that if a function $f:(a,b)\to ℝ$ is twice differentiable with $f\prime \prime \left(x\right)>0$ then $f$ is strictly convex. ( $f$ is strictly convex if $f(tx+(1-t\left)y\right)<tf\left(x\right)+(1-t)f\left(y\right)$ for all $x,y\in (a,b)$ and $t,y\in (0,1)$.
   
   

Picard and Newton iteration

1. Let $f:(0,\frac{1}{2}\pi )\to ℝ$ is given by $f\left(x\right)=\frac{1}{2}\tan x$. Estimate numerically the solution to $x=f\left(x\right)$ with $x\in (0,\frac{1}{2}\pi )$ using Picard iteration.
   
   
2. Let $f:(0,\frac{1}{2}\pi )\to ℝ$ is given by $f\left(x\right)=\frac{1}{2}\tan x$. Estimate numerically the solution to $x=f\left(x\right)$ with $x\in (0,\frac{1}{2}\pi )$ using Newton iteration (let $F\left(x\right)=x-f\left(x\right)$).
   
   
3. Show that the equation $g\left(x\right)=x^3+x-1=0$ has a solution between 0 and 1. Transform the equation to the form $x=f\left(x\right)$ for a suitable function $f:[0,1]\to [0,1]$. Use Picard iteration to find the solution to 3 decimal places. (Try $f\left(x\right)=1/(x^2+1)$).
   
   
4. Show that the equation $g\left(x\right)=x^4-4x^2-x+4=0$ has a solution between $\sqrt{3}$ and 2. Transform the equation to the form $x=f\left(x\right)$ for a suitable function $f:[\sqrt{3},2]\to [\sqrt{3},2]$. Use Picard iteration to find the solution to 3 decimal places. (Try $f\left(x\right)=\sqrt{2+\sqrt{x}}$).
   
   

Topology

1. 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,

   
   
2. 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,

   (j)   fundamental system of neighborhoods,

   (k)   continuous at $x=a$,

   (l)   continuous,

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

   (a)   topological space,

   (b)   Hausdorff,

   (b)   fundamental system of neighborhoods,

   (b)   basis,

   (c)   connected set,

   (d)   compact set,

   
   
4. Prove that $ℬ$ and is a basis of $𝒯$ if and only if $ℬ$ satisfies: if $x\in X$ then $ℬ\left(x\right)=\{B\in ℬ\hspace{0.17em}|\hspace{0.17em}x\in B\}$ is a fundamental system of neighborhoods of $x$.
   
   
5. Let $X$ and $Y$ be metric spaces. Define the topology on $X$ and $Y$. Prove that $f:X\to Y$ is continuous as a function between metric spaces if and only if $f:X\to Y$ is continuous as a function between topological spaces.
   
   
6. Define the following and give an example for each:

   (b)   filter,

   (c)   finer,

   (b)   filter base,

   (b)   neighborhood filter,

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

   (b)   Fréchet filter,

   (d)   limit of $\left(x_n\right)$ as $n\to \infty$.

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

   (c)   ultrafilter,

   (d)   quasicompact,

   (d)   Hausdorff,

   (d)   compact,

   
   
8. Let $X$ be a Hausdorff topological space and let $K$ be a compact subset of $X$. Show that $K$ is closed.
   
   
9. Let $X$ be a metric space. Show that $X$ is Hausdorff and has a countable basis.
   
   
10. Let $X$ be a metric space and let $K$ be a compact subset of $X$. Show that $K$ is closed and bounded.
    
    
11. Let $X$ be a metric space and let $E$ be a subset of $X$. Show that $E$ is compact if and only if every infinite subset of $E$ has a limit point in $E$. (What is the definition of limit point???)
    
    
12. Let $K$ be a subset of ${ℝ}^n$. Show that $K$ is compact if and only if $K$ is closed and bounded.
    
    
13. Let $X$ and $Y$ be topological spaces and let $f:X\to Y$ be a continuous function. Show that if $X$ is connected then $f\left(X\right)$ is connected.
    
    
14. Let $E\subseteq ℝ$. Show that $E$ is connected if and only if the set $E$ satisfies if $x,y\in E$ and $z\in ℝ$ and $x<z<y$ then $z\in E$.
    
    
15. (Intermediate Value Theorem) Let $f:[a,b]\to ℝ$ be a continuous function. Show that if $z\in ℝ$ and $f\left(a\right)<z<f\left(b\right)$ then there exists $c\in (a,b)$ such that $f\left(c\right)=z$.
    
    
16. Let $X$ and $Y$ be topological spaces and let $f:X\to Y$ be a continuous function. Show that if $X$ is compact then $f\left(X\right)$ is compact.
    
    
17. Let $D$ be a closed bounded subset of $ℝ$ and let $f:D\to ℝ$ be a continuous function.

    (a)   $f$ is a bounded function,

    (b)   $f$ attains its maximum and minimum on $D$,

    (a)   $f$ is uniformly continuous.