Topic 2 - The Real Numbers

Question 1

Define: Upper Bound

Answer 1

A real number that is greater than (or equal to) all of the elements in a set.

i.e. s ≤ M, ∀s ∈ S

Question 2

Define: Lower Bound

Answer 2

A real number that is less than (or equal to) all of the elements in a set.

i.e. s ≥ m, ∀s ∈ S.

Question 3

Define: Bounded

Answer 3

A set that is bounded above (contains an upper bound) and bounded below (contains a lower bound).

Question 4

Define: Supremum

Answer 4

A number T ∈ R is called the supremum of S, denoted sup(S), if
- T is an upper bound of S,
- any other upper bound M of S satisfies T ≤ M.

i.e. the smallest upper bound of a set.

Question 5

Define: Infimum

Answer 5

A number t ∈ R is called the infimum of S, denoted inf(S), if
- t is a lower bound of S,
- any other lower bound m of S satisfies t ≥ m.

i.e. the largest lower bound of a set.

Question 6

Theorem: Archimedean Postulate

Answer 6

For any real number a, there exists a natural number greater than a.

With Quantifiers: ∀a ∈ R, ∃n ∈ N s.t. a < n.

Question 7

Completeness Axiom

Answer 7

Every non-empty set bounded above has a supremum and every non-empty set bounded below has an infimum.