Solution To Problems in Real Analysis I [MAT206 Test Solution]

INTRODUCTION TO REAL ANALYSIS

Today i will be treating some problems in real analysis I, problems like the field axioms,

countable sets, infinite sets, infimum and supremum of subsets of real number, convergent and monotone

sequences. Although i have discussed on many of this topics before, you can refer back to my

previous lectures by clicking on any of the following links below.




`(1) The Monotone Convergence Sequence
(2) How To Determine if a Sequence is Monotonic[Examples and Solutions]
(3) The Convergent Sequence
(4) Solving Convergent Sequences Using Epsilon or Limit Approach
(5) Solutions to problems on converging sequences
(6) Best Approach to To Understanding Supremum, Infimum, And The Completeness Axiom 
(7) Introduction To Sequences Of Real Numbers and The Countable Sets

MAT206 TEST QUESTIONS[REAL ANALYSIS I] 2016/2017 ACADEMIC SESSION UDUSOK, SOKOTO






(1a) Define the operations of addition and multiplication on $\mathbb{Z}_5=\{0,1,2,3,4\}$
as follows:

Show that $\mathbb{Z}_5$ satisfy all the field axioms.

(1b) Define and give two examples of the following:

(i) Countable set

(ii) Infinite set

(2a) Define and give two examples of the following:

(i) Infimum of a subset of $\mathbb{R}$

(ii) Supremum of a subset of $\mathbb{R}$

(2b) Let $S$ be a subset of real number which is bounded.

Show that the following equation holds:

$\inf(-S)=-\sup{S}$

(3a) Define and give two examples of the following:

(i) Convergent sequence

(ii) Monotone sequence

(3b) For each of the following sequence, determine whether it converges and if it converges,

guess your limit and prove your guess

(i) $\{\sqrt{n^2+1}-n\}$

(ii) $\{\sqrt{4n^2+n}-2n\}$


Question 1

(a) Define the operations of addition and multiplication on $\mathbb{Z}_5=\{0,1,2,3,4\}$ as follows:

Show that $\mathbb{Z}_5$ satisfy all the field axioms.

Solution

To proof that $\mathbb{Z}_5$ is a field, it suffices to show that it satisfies all axioms

of field structure using the given table

We start with the addition property:

Some books do add the closure law as part of the field axioms, and of course it is part of

it because all other additive rules are more or less generalizations of the closure law.

It states that for any two element in $\mathbb{Z}_5$ define an operation of addition on it,

then the result will be in ${\mathbb{Z}_5}$

(i) for any $1,2\in{\mathbb{Z}_5}$ then $1+2=2+1=3\in{\mathbb{Z}_5}$, it is cummutative

(ii) for any $1,2,3\in{\mathbb{Z}_5}$ then $(1+2)+3=1+(2+3)=1\in{\mathbb{Z}_5}$, it is

 associative

(iii) Existence of an additive identity element $0\in{\mathbb{Z}_5}$ such that

$3+0=0+3=3\in{\mathbb{Z}_5}$

(iv) Existence of an additive inverse element such that for any $2\in{\mathbb{Z}_5}$ there

exists a $-2$ such that $2+(-2)=0\in{\mathbb{Z}_5}$

Multplicative property

(v) for any $1,2\in{\mathbb{Z}_5}$ then $1(2)=2(1)=2\in{\mathbb{Z}_5}$,

the multiplication is cummutative

(vi) for any $1,2,3\in{\mathbb{Z}_5}$ then $3(1+2)=3(1)+3(2)=4\in{\mathbb{Z}_5}$,

it is distributive

(vii) Existence of a multiplicative identity element $1\in{\mathbb{Z}_5}$ such that $3\times{1}=3\in{\mathbb{Z}_5}$

(viii) Existence of a multiplicative inverse element such that for any

$2\in{\mathbb{Z}_5}$ there exists a $2^{-1}$ such that

$2\times{2^{-1}}=1\in{\mathbb{Z}_5}$

Hence we conclude that since $\mathbb{Z}_5$ satisfies all field axioms, hence it a field.

Question 1b

Define and give two examples of the following:

(i) Countable set

Solution

Countable Set: A set $X$ is said to be countable if there exists a one-one correspondence

between $X$ and the set of natural numbers.

Examples

(i) Every finite set is countable

(ii) The set $\mathbb{Z}$ of integers is also countable

Question 1b

(ii) Infinite set

Solution

Infinite Set: A set $X$ is said to be infinite if there exists a one-one correspondence

between $X$ and atleast one of its proper subsets, or if there exists a proper subset of

$X$, say $A$, $A\subseteq{X}$ such that $n(A)=n(X)$.

Examples

(i) The set $\mathbb{Z}$ of integers is infinite.

(ii) The set $S:=(-1,1)$ of open interval is infinite set.

Question 2a

Define and give two examples of the following:

(i) Infimum of a subset of $\mathbb{R}$

Solution

(i) Infimum of a subset of $\mathbb{R}$: This is defined as, Let $S$ be a subset of real

number bounded below by $\alpha_0$ then $\alpha_0$.

Then $\alpha_0$ is an infimum if it satisfies two conditions:

(i) $\alpha_0$ must be a lower bound for $S$

(ii) For all $\epsilon>0$, the number $\alpha_0+\epsilon$ is not a lower bound for $S$.

                                                   Or

Let $S$ be a subset of a real number that is bounded above and nonempty, if $m$ is the

greatest of all the lower bounds of $S$, then $m$ is said to be the infimum of the set

$S$.

Examples

(i) If $a<b$, then $b=\inf[a,b]=\inf(a,b]$

(ii) If $X=\{1,2,3\}$ then $\inf(X)=1$

Question 2a

(ii) Supremum of a subset of $\mathbb{R}$
Solution

Supremum of a subset of $\mathbb{R}$ is defined as, let $S$ be a subset of a real number

that is bounded above, the supremum or least upper bound say $\beta_0$ is a real number

satisfying two conditions:

(i)  $\beta_0$ is an upper bound for $S$.

(ii) if $s\leq{\beta}$ for all $s\in{S}$ then $\beta_0\leq{\beta}$
               
                                            Or

We can also define it as let $S$ be a set of real number that is bounded above and

nonempty. If $M$ is the supremum or least of all upper bound, then $S$ is said to be the

supremum of $S$.

Example

(i) If $a<b$ and $b=\sup[a,b]=\sup[a,b)$

(ii) Let $X=\{1,2,3\}$ then $\sup(X)=3$.

Question 2b 

Let $S$ be a subset of real number which is bounded. Show that the

following equation holds:

$\inf(-S)=-\sup{S}$

Solution

We are to proof that $\inf(-S)=-\sup{S}$, it suffices to show that

$\inf{S}=-\sup{-S}$, we let $\alpha=\inf{S}$. then

(i) $\alpha\leq{s}$ for all $s\in{S}$

(ii) for all $\epsilon>0$ there exists $s_0\in{S}$ such that

$\alpha\leq{s_0}<\alpha+\epsilon$, conditions (i) and (ii) imply

(i*) $-s\leq{-\alpha}$ for all $-s\in{-S}$

(ii*) for all $\epsilon>0$ there exists $-s_0\in{-S}$ such that

$(-\alpha)-\epsilon<-s_0\leq{-\alpha}$.

conditions (i) and (ii) imply

$-\alpha=\sup(-S)$, i.e $-\inf{S}=\sup(-S)$. Which proves the statement.

Question 3a

Define and give two examples of the following:

(i) Convergent sequence

Solution

A sequence $\{a_n\}^\infty_{n=1}$ of real number is said to converge to a real number $L$

if and only if for each $\epsilon>0$, there exists a natural number $n(\epsilon)$ such that

$|a_n-L|<\epsilon$ for all $n\geq{n(\epsilon)}$.

Examples

(i) The sequence defined by $\{b_n\}^\infty_{n=1}=\{(-1)^n\}^\infty_{n=1}$ is the infinite

sequence $\{-1,1,-1,1,...\}$, which is convergent.

(ii) The sequence $\{a_n\}^\infty_{n=1}=\{3\}^\infty_{n=1}$ is the constant sequence

$\{3,3,3,...\}$ whose set of values is the singleton $\{3\}$,

and it is thus a convergent sequence.

Question 3a

(ii) Monotone sequence

Solution
If $\{a_n\}$ is a sequence of real numbers then $\{a_n\}$ is monotonic
if and only if the following holds for all $n\in{\mathbb{N}}$:

(i) Monotone non-decreasing if $a_{n+1}\geq{a_n}$

(ii) Strictly monotone increasing if $a_{n+1}>a_n$

(iii) Monotone non-increasing if $a_{n+1}\leq{a_n}$

(iv) Strictly monotone decreasing if $a_{n+1}<a_n$

Examples
(i) The sequence $\{a_n\}=\{1-\frac{1}{n}\}$ is a strictly monotone

increasing sequence if $n\geq{1}$

(ii) The sequence $\{a_n\}=\{\frac{1}{n^2}\}$ for all $n\geq{1}$ is a

strictly monotone decreasing sequence.

Question 3b

For each of the following sequence, determine whether it converges and if it converges,

guess your limit and prove your guess

(i) $\{\sqrt{n^2+1}-n\}$

Solution

We are to ascertain if the sequence is convergent and hence guess a limit for it.

Let our guess be $\frac{1}{2}$, so that we proof the sequence

$\{\sqrt{n^2+1}-n\}=\frac{1}{2}$,

our best option here is to rationalise the sequence in radical form.

$\{\sqrt{n^2+1}-n\}=\frac{(\sqrt{n^2+1}-n)(\sqrt{n^2+1}+n)}{\sqrt{n^2+1}+n}$ expand the

numerator.

$=\frac{1}{\sqrt{n^2+1}+n}$

Now we take the limit of $n$ to infinity

$\lim_{n\rightarrow\infty}=\frac{1}{\sqrt{n^2+1}+n}=\frac{1}{2}$

Hence since our limit exists at $n\rightarrow\infty$ then the sequence is convergent and

hence our limit exists.

Question 3b

(ii) $\{\sqrt{4n^2+n}-2n\}$
Solution

We use the same method as above, all we need to do is assume the limit of the sequence to

be $\frac{1}{4}$, so we proof our guess and determine if the sequence is truly convergent.

$\{\sqrt{4n^2+n}-2n\}=\frac{(\sqrt{4n^2+n}-2n)(\sqrt{4n^2+n}+2n)}{\sqrt{4n^2+n}+2n}$

Expand the numerator

$=\frac{n}{\sqrt{4n^2+n}-2n}$

factor out $n^2$ from the denominator so that we get

$=\frac{1}{\sqrt{4+\frac{1}{n}}-2}$ 

Now take the limit of $n\rightarrow\infty$

$\lim_{n\rightarrow\infty}\frac{1}{\sqrt{4}+2}=\frac{1}{4}$

Hence, our limit exists at $n\rightarrow\infty$, so we conclude that our sequence is convergent.