This document contains solutions to exercises on real analysis. It includes: 1) Proving various properties relating to if a is a real number. 2) Solving equations by justifying each step with appropriate theorems or properties. 3) Proving that if a satisfies a^2 = 2, then either a = √2 or a = -√2. 4) Showing there does not exist a rational number whose square is 2 by reducing it to a contradiction.

1. INTRODUCTION TO REAL ANALYSIS 1
INDIVIDUAL TASK
EXERCISES 2.1
By:
Muhammad Nur Chalim
4101414101
MATHEMATICS DEPARTMENT
MATHEMATICS AND NATURAL SCIENCES FACULTY
SEMARANG STATE UNIVERSITY
2016

2. EXERCISES 2.1
Problem
1. If , prove the following
a. If , then
b. ( )
c. ( )
d. ( )( )
2. Prove that if , then
a. ( ) ( ) ( )
b. ( ) ( )
c.
( )
( )
d. ( )
( )
3. Solve the following equations, justifying each step by referring to an appropriate property
or theorem
a.
b.
c.
d. ( )( )
4. If satisfies , prove that either or
5. If and , show that
( )
( )( )
6. Use the argument in the proof of Theorem 2.1.4 to show that there does not exist a
rational number such that
Solution
1. Given
a. If , then
(A3)
( ) (A4)
( ) (A2)

3. (Hypothesis)
(A3)
( )
b. ( ) ( ) (A3)
( ) ( ) ( ) (A4)
( ) ( ( ) ( )) (A2)
( ) (A4)
( ) (A3)
( )
c. ( ) ( ) (A3)
( ) ( ) ( ) (A4)
( ) ( ( ) ) (A2)
( ) ( ( ) ) (M3)
( ) ( ( )) (D)
( ) (A4)
( ) (Theorem 2.1.2 c)
( ) (A3)
( )
d. Based on (1c) we have ( )( ) ( ) and based on (1b) we get ( ) .
Thus, ( )( ) ( )
( )
2.

4. a. ( ) ( )( ) (from 1c)
( ) ( ) ( ) (D)
( ) ( ) ( ) (from 1c)
( )
b. ( ) ( ) ( ) ( ) (from 1c)
( ) ( ) ( )( ) (M1)
( ) ( ) (from1d)
( ) ( ) (M3)
( )
c. (M3)
( ) (M4)
( ( ( ))) (M4)
( ( )) ( ( )) (M2)
( ( )) (M4)
( ) (M3)
( )
d. ( ) ( ) ( ) (from 1c)
( ) ( ) ( ) (since )
( ) (( ) ) (M2)
( ) ( ) (from 1c)
( )
( )
(since )
( )

5. 3. Solve the equations
a.
( ) ( ) (Add to both sides)
( ( )) (A2)
(A4)
(A3)
(Multiply by to both sides)
( ) (M2)
(M4)
(M3)
b.
( ) ( ) ( add by both sides)
( ) (A4)
( ) (D)
(Theorem 2.1.3b)
(add by both sides of )
(A4)
(A3)
c.

6. ( ) ( ) (Add by to both sides)
(A4)
( )( ) (Factorization)
(Theorem 2.1.3b)
( ) ( )
(A4)
(A3)
d. ( )( )
(Theorem 2.1.3b)
( ) ( )
(A4)
(A3)
4. If satisfies , prove that either or . It suffices to assume
and prove that .
(M3)
( ) (M4)
( ) (M2)
(Hypothesis)
(M4)
( )
5. If , then we get
(M3)

7. ( ) (M3)
( ) (M2)
( ) (M4)
( ) (M1)
( ) (M2)
( ) (M4)
(M3)
( )
6. Use the argument in the proof of Theorem 2.1.4 to show that there does not exist a
rational number such that
Proof:
Suppose, on the contrary, that and are integers such that ( )
. We may assume that and are positive and have no common integer
factors other than or the other word that ( ) , since , we see that
is even. This implies that is also even. Therefore, since and do not have a common
factor, then must be and odd natural number.
Since is even, then for some , and hence , so that
. Therefore, is even. Since is even and is an odd then is even.
This implies that is also even.
Since the hypothesis that ( ) leads to the contradictory conclusion that is
both even and odd, it must be false
Thus, there does not exist a rational number such that