2 The real numbers

Question 1

2.2 Order Axioms

There are 5

Answer 1

[(A10)] For any a, b ∈ R, either a ≤ b or b ≤ a.
[(A11)] For all a, b ∈ R, if a ≤ b and b ≤ a, then a = b.
[(A12)] For all a, b, c ∈ R, if a ≤ b and b ≤ c, then a ≤ c.
[(A13)] For all a, b, c ∈ R, if a ≤ b, then a + c ≤ b + c.
[(A14)] For all a, b, c ∈ R, if a ≤ b and c ≥ 0, then ac ≤ bc.

Question 2

Definition 2.1:
Upper bound

Answer 2

Consider a set S ⊆ R.
* A number M ∈ R is said to be an upper bound of S if s ≤ M for all s ∈ S.

Question 3

Definition 2.1:
Lower bound

Answer 3

Consider a set S ⊆ R.
* A number m ∈ R is said to be a lower bound of S if s ≥ m for all s ∈ S.

Question 4

Definition 2.1:
Bounded set

Answer 4

Consider a set S ⊆ R.
* The set S is called bounded above if it has an upper bound and bounded below if it has a lower bound. We say that S is bounded if it is bounded above and below.

Question 5

Definition 2.2:
Supremum

Answer 5

Consider a set S ⊆ R.
* A number T ∈ R is called supremum or least upper bound of S if T is an upper bound of S
and any other upper bound M of S satisfies T ≤ M.

Question 6

Definition 2.2:
Infimum

Answer 6

Consider a set S ⊆ R.
* A number t ∈ R is called infimum or greatest lower bound of S if t is a lower bound of S and
any other lower bound m of S satisfies t ≥ m.

Question 7

[(A15)] The completeness axiom

Answer 7

Every non-empty set of real numbers that is bounded above has a supremum. Every non-empty set of real numbers that is bounded below has an infimum.

The supremum and infimum of a set (if they exist) are unique.

Question 8

Theorem 2.1

Answer 8

Let S ⊆ R be a non-empty set.
* Suppose that S is bounded above. Then for any ε>0 there exists s∈S such that s>supS−ε.
* Suppose that S is bounded below. Then for any ε>0 there exists s∈S such that s<infS+ε.

Question 9

Theorem 2.2:
Archimedean postulate

Answer 9

For any real number a∈R there exists n∈N with a<n.