Chapter 1 - Real numbers

Question 1

Absolute value function definition

Answer 1

The absolute value function is defined by
|x |= x for x>0, -x for x<0
for all a,b is any real number

|ab| = |a||b |
|a+b| < | a| + | b|

Question 2

The triangle inequality

Answer 2

|a-b|=|a-c+c-b|=|(c-a)+(c-b) |< |a-c| + |c-b|

Question 3

The archimedean axiom

Answer 3

for every x ∈ IR there exists a natural number n ∈ N such that n>x

Question 4

Definition 1.3 - bounds

Answer 4

A set A, A ≤ IR (subset) is bounded above if there exists a number m ∈ IR such that a≤m. The number m is an upper bound for A.

Question 5

Definition 1.4 - supremum

Answer 5

A number S ∈ IR is the least upperbound /supremum for a set A is a subset or R if
1. S is an upper bound of A
2. M is any upper bound of A, S ≤ M

s = supA

Question 6

Axiom of completeness

Answer 6

every non empty set of real numbers that is bounded above has a supremum

Question 7

Theorem - Nested interval property

Answer 7

For each n ∈ natural number, we are given a closed interval,
In = [an,bn] ={x∈R | an<x<bn}
Assume each In contains In+1, then I1 ⊃I2⊃I3 … has a non empty intersection, there exists an element y∈R such that y ∈ In for all n ∈ R